probability.tex

\begin{enumerate}
\item Let A and B be two events such that $P(A) = \frac{5}{8}$, $P(B) = \frac{1}{2}$ and  $P(A|B) = \frac{3}{4}$. Find the value of $P(B|A)$.
\item Two balls are drawn at random from a bag containing 2 red balls and 3 blue balls, without replacement. Let the variables X denotes the number of red balls. Find the probabillity distribution of X.
\item A card from a pack of 52 playing cards is lost. From the remaining cards, 2 cards are drawn at random without replacement, and are found to be both aces. Find the probability that lost card being an ace.
\item Probabilities of A and B solving a specific problem are $\frac{2}{3}$ and $\frac{3}{5},$ respectively. If both of them try independently to solve the problem, then find the probability that the problem is  solved.
\item A pair of dice is thrown. It is given that the sum of numbers  appearing on both dice is an even number. Find the probability that the number apprearing on at least one die is 3.
\item At the start of a cricket match, a coin is tossed and the team winning the toss has the opportunity to choose to bat or bowl. such a coin is unbaised with equal probabilities of getting head and tail\figref{fig:coin1} .
\begin{figure}[!ht]
\centering
\includegraphics[width=\columnwidth]{figs/coin}
\caption{Toss before the match}
\label{fig:coin1}
\end{figure}
\\ Based on the above information, answer the following question:
\begin{enumerate}
\item If such a coin is tossed 2 times, then find the probability distibution of numbers of tails.
\item Find the probability of getting at least one head in three tosses of such a coin.
\end{enumerate}
\item Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of spade cards.
\item A pair of dice is thrown and the sum of the numbers appearing on the dice is observed to be 7. Find the probability that the number 5 has appeared on at least one die.
\item The probability that A hits the target is $\frac{1}{3}$ and the probability that B hits it, is $\frac{2}{5}.$ If both try to hit the target independently, find the probabillity that the target is hit. 
\item A shopkeeper sells three types of flower seeds A$_1$ , A$_1$ , A$_3$. They are sold in the form of a mixture, where the proportions of these seeds are  4 : 4 : 2, respectively. The germinaton rates of the three types of seeds are $45\%,$ $60\%$ and $35\%$ respectively\figref{fig:flowers11}.
\begin{figure}[!ht]
\centering                                  \includegraphics[width=\columnwidth]{figs/flowers}                                     
\caption{Three types of flowers}            
\label{fig:flowers11}                       
\end{figure}
\\ Based on  the above information :
\begin{enumerate}
\item  Calculate the probability that a randomly chosen seed will germinate.
\item  Calculate the probability  that the seed is of type $A_2$, given that a randomly choosen seed germinates.
\end{enumerate}
\item Three friends A, B and C got their photograph clicked. Find the probability that B is standing at the central position, given that A is standing at the left corner.
\item In a game of Archery, each ring of the Archery target is valued. The centremost ring is worth 10 points and rest of the rings are alloted points 9 to 1 in sequential order moving outwards.Archer A is likely to earn 10 points with a probability of 0.8 and Archer B is likely the earn 10 points with a probability of 0.9\figref{fig:archery3}.
\begin{figure}[!ht]                     
\centering
\includegraphics[width=\columnwidth]{figs/archery}
\caption{centermost ring}                   
\label{fig:archery3}                        
\end{figure}
\\ Based on the above innformation, answer the following questions :
\begin{enumerate}
\item exactly one of them earns 10 points .
\item both of them earn 10 point.
\end{enumerate}
\item Event A and B are such that \begin{align} P(A) = \frac{1}{2},  P(B) = \frac{7}{12}\end{align} and \begin{align} P(\bar{A}\cup \bar{B}) = \frac{1}{4} \end{align}
Find whether the events  A and B are independent or not.
\item A box $B_1$ contain 1 white ball  and 3 red balls. Another box $B_2$ contains 2 white balls and 3 red balls. If one ball is drawn at random from each of the boxes $B_1$ and $B_2$, then find the probability that the two balls drawn are of the same colour.
\item Let X be random variable which assumes values $x_1$, $x_2$, $x_3$, $x_4$  such that\begin{align} 2P(X = x_1) = 3P (X = x_2) = P ( X = x_3) = 5P (X = x_4).\end{align}
\\ Find the probability distribution of X.
\item There are two boxes, namely box-I and box-II. Box-I contains  3 red and 6 black balls. Box-II contains 5 red and 5 black balls. One of the two boxes , is selected at random and a ball is drawn at random. The ball drawn is found to be red. Find the probability that this red ball comes out from box-II.
\item In a toss of three different coins, find the probability of comming up of three heads, if it is known that at least one head comes up.
\item A laboratory blood text is $98\%$ effective  in detecting a certain disease when it is fact, present. However, the text also yeilds a false positive result for $0.4\%$ of the healthy person tested. From a large population, it is given that $0.2\%$ of the population actually has the diseases.
\\Based on the above, answer the following questtion : 
\begin{enumerate}
\item one person, from the population, is taken at random and given the test. Find the probabiliy of his getting a positive test result.
\item what is the probability that the person actually has the disease, given that his test result is positive ?
\end{enumerate}
\item Two cards are drawn from a well-shuffled pack of playing cards one-by-one with replacement. The probability that the first card is a king and the second card is a queen is 
\begin{enumerate}
\item $\frac{1}{13} + \frac{1}{13}$
\item $ \frac{1}{13} \times \frac{4}{51}$
\item $\frac{4}{52} \times \frac{3}{51}$
\item $\frac{1}{13} \times \frac{1}{13}$
\end{enumerate}
\item For two events A and B if P(A) = $\frac{4}{10}, P{B} = \frac{8}{10}$ and $P(B|A)$ = $\frac{6}{10}$ then find $P( A \cup B).$
\item Bag I contain 4 red and 3 black balls. Bag II contains 3 red and 5 black balls. One of two bags is selected at random and a ball is drawn from the bag, which if found to be red. Find the probability that the ball is drawn from bag II.
\item Two cards are drawn successively without replacement from a well-shuffled pack of 52 cards. Find the probability distribution of the number of aces and hence find its mean.
\item The probability of solving a specific question independently by A and B are $\frac{1}{3}$ and $\frac{1}{5}$ respectively . If both try to solve the question independently, the probability that the question is solved is 
\begin{enumerate}
\item $\frac{7}{15}$
\item $\frac{8}{15}$
\item $\frac{2}{15}$
\item $\frac{14}{15}$
\end{enumerate}
\item A card is picked at random from a pack of 52 playing cards. Given that the picked up card is a queen, the probability of it being a queen of spades is \underline{\hspace{1cm}}.
\item A bag contains 19 tickets, numbered 1 to 19. A ticket is drawn at random and then another ticket is drawn without replacing the first one in the bag. Find the probability distribution of the number of even numbers on the ticket.
\item Find the probability distribution of the numbers of successes in two tosses of a die, when a success is defined as number greater than 5.
\item Ten cartoons are taken at random from an automatic packing machine. The mean net weight of the ten carton is 11.8 kg and standard deviation is 0.15 kg. Does the sample mean differ significantly from the intended mean of 12 kg ?
[Given that for d.f. = 9, $t_{0.05}$ = 2.26]
\item A Coin is tossed twice. The following table\ref{tab: Number of tails} shows the probability distribution of numbers of tails:
\begin{table}[!ht]
\input{./2022/tablep.tex}	
\caption{Table shows the probability distribution of numbers of tails \label{tab: Number of tails}}
\end{table}
\begin{enumerate}
\item Find the value of $K$.
\item Is the coin tossed biased or unbaised?
Justify your answer.
\end{enumerate}
\item If X is a random variable with probability distribution as given below \ref{tab:probability distribution}:
\begin{table}[!ht]
\input{2022/tableb.tex}
\caption{table shows the proability distribution \label{tab:probability distribution}}
\end{table}
\newline The value of K and the mean of the distribution respectively are 
\begin{enumerate}
\item $\frac{1}{7}, 1$
\item $\frac{1}{6}, 2$
\item $\frac{1}{6}, 1$
\item $1, \frac{1}{6}$
\end{enumerate}
\item The random variable X has a probability function P($x$) as defined below, where K is some number :
\\ \begin{align}P(X)=\begin{cases} K, & \text{if }  x=0 \\ 2K, & \text{if } x=1\\ 3K, & \text{if } x=2\\ 0, & \text{otherwise  } \end{cases}\end{align}
\\ Find:
\begin{enumerate}
\item The value of $K$.
\item $P(X<2),P(X \le 2), P(X \ge 2)$.
\end{enumerate}
\item Two rotten apples are mixed with 8 fresh apples. Find the probability distribution of number of rotten apples, if two apples are drawn at  random, one-by-one without replacement.

\item A die is thrown twice. What is the probability that 
\begin{enumerate}[label=(\roman*)]
 \item $5$ will come up at least once, and 
 \item $5$ will not come up either time ? 
\end{enumerate}

\item Let $A$ and $B$ be two events such that $P(A)=\frac{5}{8}$, $P(B)=\frac{1}{2}$ and $P(A/B)=\frac{3}{4}$. Find the value of $P(B/A)$.

\item Two balls are drawn at random from a bag containing $2$ red balls and $3$ blue balls, without replacement. Let the variable $X$ denotes the number of red balls. Find the probability distribution of $X$.

\item A card from a pack of $52$ playing cards is lost. From the remaining cards, $2$ cards are drawn at random without replacement, and are found to be both aces. Find the probability that lost card being an ace.

\item Probabilities of $A$ and $B$ solving a specific problem are $\frac{2}{3}$ and $\frac{3}{5}$, respectively. If both of them try independently to solve the problem, then 
find the probability that the problem is solved.

\item A pair of dice is thrown. It is given that the sum of numbers appearing on both dice is an even number. Find the probability that the number appearing on at least one die is $3$.

\item In \figref{fig:fig1.png},At the start of a cricket match, a coin is tossed and the team winning the 
toss has the opportunity to choose to bat or bowl. Such a coin is unbiased 
with equal probabilities of getting head and tail.

\begin{figure}[H]
        \centering
        \includegraphics[width=\columnwidth]{./figs/Screenshot (19).png}
        \caption{Tossing a coin}
        \label{fig:fig1.png}
    \end{figure}

Based on the above information, answer the following questions :
\begin{enumerate}[label=(\alph*)]
 \item  If such a coin is tossed $2$ times, then find the probability 
distribution of number of tails.
 
 \item Find the probability of getting at least one head in three tosses of 
such a coin. 
\end{enumerate}

\item Two cards are drawn successively with replacement from a well shuffled pack of $52$ cards. Find the probability distribution of the number of spade cards.

\item A pair of dice is thrown and the sum of the numbers appearing on the dice is observed to be $7$. Find the probability that the number $5$ has appeared on atleast one die.

\item In \figref{fig:fig2.png}, A shopkeeper sells three types of flower seeds $A1$, $A2$, $A3$. They are sold in the form of a mixture, where the proportions of these seeds are $4:4:2$, respectively. The germination rates of the three types of seeds are $45\%$, $60\%$ and $35\%$ respectively.

\begin{figure}[H]
        \centering
        \includegraphics[width=\columnwidth]{./figs/Screenshot (23).png}
        \caption{Three Types of Flower Seeds}
        \label{fig:fig2.png}
    \end{figure}

    Based on the above information:
    
    \begin{enumerate}[label=(\alph*)]
    
 \item Calculate the probability that a randomly chosen seed will germinate;
 
 \item Calculate the probability that the seed is of type $A2$, given that a randomly chosen seed germinates.

\end{enumerate}

\item Three friends $A$, $B$ and $C$ got their photograph clicked. Find the 
probability that $B$ is standing at the central position, given that $A$ is 
standing at the left corner. 

\item In \figref{fig:fig3.png} A coin is tossed twice. The following table shows the probability 
distribution of number of tails :
\begin{figure}[H]
        \centering
        \includegraphics[width=\columnwidth]{./figs/Screenshot (28).png}
        \caption{Probability Distribution of number of tails}
        \label{fig:fig3.png}
    \end{figure}

    \begin{enumerate}[label=(\alph*)]
    
 \item  Find the value of $K$. 
 
 \item  Is the coin tossed biased or unbiased ? Justify your answer.

\end{enumerate}

\item In \figref{fig:fig4.png} In a game of Archery, each ring of the Archery target is valued. The 
centre most ring is worth $10$ points and rest of the rings are allotted 
points $9$ to $1$ in sequential order moving outwards.

Archer A is likely to earn $10$ points with a probability of $0·8$ and Archer $B$ 
is likely the earn $10$ points with a probability of $0·9$.

\begin{figure}[H]
        \centering
        \includegraphics[width=\columnwidth]{./figs/Screenshot (26).png}
        \caption{Ring of the Archery Target}
        \label{fig:fig4.png}
    \end{figure}

Based on the above information, answer the following questions : 
If both of them hit the Archery target, then find the probability that 

\begin{enumerate}[label=(\alph*)]
    
 \item  exactly one of them earns $10$ points.
 
 \item  both of them earn $10$ points.

\end{enumerate}


\item 
\begin{enumerate}[label=(\alph*)]
    
 \item  Events $A$ and $B$ are such that
 P(A) =  $\frac{1}{2}$ , P(B) =  $\frac{7}{12}$  and $ P( \overline{A}  \cup  \overline{B} )= \frac{1}{4}$ Find whether the events $A$ and $B$ are independent or not.
 
 \item  A box $B_{1}$ contains $1$ white ball and $3$ red balls.Another box $B_{2}$ contains $2$ white balls and $3$ red balls.If one ball is drawn at random from each of the boxes $B_{1}$ and $B_{2}$ then find the probability that the two balls drawn are of the same colour.
 
\end{enumerate}

 \item There are two boxes, namely box-I and box-II. Box-I contains $3$ red and $6$ black balls. Box-II contains $5$ red and $5$ black balls. One of the two boxes, is selected at random and a ball is drawn at random. The ball drawn is found to be red. Find the probability that this red ball comes out from box-II.

\item In a toss of three different coins, find the probability of coming up of three heads, if it is known that at least one head comes up.

\item Two rotten apples are mixed with $8$ fresh apples. Find the probability distribution of number of rotten apples, if two apples are drawn at random, one-by-one without replacement.

\item A laboratory blood test is $98\%$ effective in detecting a certain 
disease when it is in fact, present. However, the test also yields 
a false positive result for $0·4\%$ of the healthy person tested. 
From a large population, it is given that 0·2$\%$ of the population 
actually has the disease. 
Based on the above, answer the following questions : 

  \begin{enumerate}[label=(\alph*)]
    
 \item One person, from the population, is taken at random and 
given the test. Find the probability of his getting a 
positive test result.  
 
 \item  What is the probability that the person actually has the 
disease, given that his test result is positive ?

\end{enumerate}

\item Two cards are drawn from a well-shuffled pack of playing 
cards one-by-one with replacement. The probability that the 
first card is a king and the second card is a queen is 

\begin{enumerate}[label=(\alph*)]
    
 \item $\frac{1}{13} + \frac{1}{13}$
 
 \item $\frac{1}{13} \times \frac{4}{51}$

 \item $\frac{4}{52} \times \frac{3}{51}$
 
 \item $\frac{1}{13} \times \frac{1}{13}$ 

\end{enumerate}

\item In \figref{fig:fig5.png} If $X$ is a random variable with probability distribution as given 
below :
\begin{figure}[H]
        \centering
        \includegraphics[width=\columnwidth]{./figs/Screenshot (32).png}
        \caption{Probability Distribution}
        \label{fig:fig5.png}
    \end{figure}

The value of $k$ and the mean of the distribution respectively 
are

 \begin{enumerate}[label=(\alph*)]
    
 \item  $\frac{1}{7}$,1 
 
 \item  $\frac{1}{6}$,2

 \item  $\frac{1}{6}$,1

 \item  $\frac{1}{6}$

\end{enumerate}


\item For two events $A$ and $B$ if P(A) = $\frac{4}{10}$, P(B) = $\frac{8}{10}$ and 
$ P(B \mid A)$=$\frac{6}{10}$, then find $ P(A \cup B)$.

\item Bag I contains $4$ red and $3$ black balls. Bag II contains $3$ red 
and $5$ black balls. One of the two bags is selected at random 
and a ball is drawn from the bag, which is found to be red. 
Find the probability that the ball is drawn from Bag II.

\item Two cards are drawn successively without replacement from a 
well-shuffled pack of $52$ cards. Find the probability 
distribution of the number of aces and hence find its mean.
\newpage

\item The probability of solving a specific question independently by $A$ and $B$ 
are $\frac{1}{3}$ and $\frac{1}{5}$ respectively. If both try to solve the question independently, 
the probability that the question is solved is 

\begin{enumerate}[label=(\alph*)]
    
 \item  $\frac{7}{15}$
 
 \item  $\frac{8}{15}$
 
 \item  $\frac{2}{15}$
 
 \item  $\frac{14}{15}$

\end{enumerate}

\item A card is picked at random from a pack of $52$ playing cards. Given that 
the picked up card is a queen, the probability of it being a queen of 
spades is ?

\item A bag contains $19$ tickets, numbered $1$ to $19$. A ticket is drawn at random 
and then another ticket is drawn without replacing the first one in the 
bag. Find the probability distribution of the number of even numbers on 
the ticket.

\item Find the probability distribution of the number of successes in two tosses 
of a die, when a success is defined as ‘‘number greater than $5$’’.

\item The random variable $X$ has a probability function $P(x)$ as defined below, 
where $k$ is some number :

\begin{align}
    p(x) = \begin{cases}
        k, & \text{if } x = 0, \\
        2k, & \text{if } x = 1, \\
        3k, & \text{if } x = 2, \\
        0, & \text{otherwise.}
    \end{cases}
\end{align}

Find :
\begin{enumerate}[label=(\roman*)]
 \item The value of $k$
 
 \item $P(X < 2)$, $P(X \leq 2)$, $P(X\ \geq 2)$
 
 \end{enumerate}

\item Consider the following hypothesis :

\begin {align}
H0 : \mu =  35\\
H1 : \mu \neq 35
\end{align}
A sample of $81$ items is taken whose mean is $37·5$ and the standard deviation is $5$. Test the hypothesis at $5\%$ level of significance.

[Given : Critical value of $Z$ for a two-tailed test at $5\%$ level of significance is $1.96$]

\item In \figref{fig:fig6.png} Fit a straight line trend by the method of least squares and find the trend 
value for the year $2008$ for the following data :

\begin{figure}[H]
        \centering
        \includegraphics[width=\columnwidth]{./figs/Screenshot (37).png}
        \caption{Years and Production}
        \label{fig:fig6.png}
    \end{figure}
\end{enumerate}