Probability Distribution

Q. A person throws two fair dice. He wins Rs. $15$ for throwing a doublet (same numbers on the two dice), wins Rs. $12$ when the throw results in the sum of $9$, and loses Rs. $6$ for any other outcome on the throw. Then the expected gain/loss (in Rs.) of the person is :

1. $2$ gain
2. $\frac{1}{2}$ loss
3. $\frac{1}{4}$ loss
4. $\frac{1}{2}$ gain

Q.

If$A,B,C$ are events such that $P\left(A\right)=0.3,P\left(B\right)=0.4,P\left(C\right)=0.8,P(A\cap B)=0.08,P(A\cap C)=0.28,P(A\cap B\cap C)=0.09.IfP(A\cup B\cup C)\ge 0.75$, then find the range of $x=P(B\cap C)$ lies in the interval

1. $0.23\le x\le 0.48$

2. $0.23\le x\le 0.47$

3. $0.22\le x\le 0.48$

4. None of these

Q. If the probability of hitting a target by a shooter, in any shot, is $\frac{1}{3}$, then the minimum number of independent shots at the target required by him so that the probability of hitting the target at least once is greater than $\frac{5}{6}$, is :

1. $3$
2. $4$
3. $5$
4. $6$

Q. If, getting a number greater than $4$ on a fair die is considered a success, then the variance of the distribution of success on tossing a die five times is

1. $\frac{5}{4}$
2. $\frac{10}{9}$
3. $\frac{2}{3}$
4. $\frac{1}{3}$

Q. A fair die is tossed repeatedly until a six is obtained. Let X denote the number of tosses required. The probability that $X\ge 3$ equals

1. $\frac{125}{36}$
2. $\frac{25}{36}$
3. $\frac{5}{36}$
4. $\frac{25}{216}$

Q. A biased coin with probability of heads $p(0<p<1)$, is tossed until a head appears for the first time. If the probability that the number of tosses required is even is $\frac{2}{5}$, then $3p$ is equal to

Q. In a game, a man wins Rs. $1000$ if he gets an even number $\ge 4$ on a fair die and loses Rs. $200$ for getting any other number on the die. If he decides to throw the die until he wins or maximum of three times, then his expected gain/loss (in Rupees) is

1. $\frac{3800}{9}\text{loss}$
2. $\frac{3800}{9}\text{gain}$
3. $0$
4. $400\text{gain}$

Q. A random variable $X$ has probability distribution

$\begin{array}{ccccccccc}X& 1& 2& 3& 4& 5& 6& 7& 8\\ P\left(X\right)& 0.13& 0.22& 0.12& 0.21& 0.13& 0.08& 0.06& 0.05\end{array}$

If events are $E=\left\{x\text{is an odd number}\right\},$$F=\left\{x\text{is divisible by}3\right\}$ and $G=\left\{x\text{is less than}7\right\}$, then the value of $P(E\cup (F\cap G\left)\right)$ is

1. $0.87$
2. $0.77$
3. $0.52$
4. $0.82$

Q. A bag contains $30$ white balls and $10$ red balls. $16$ are drawn one by one randomly from the bag with replacement. If $X$ be the number of white balls drawn, then $\left(\frac{\text{mean of X}}{\text{standard deviation of X}}\right)$ is equal to :

1. $3\sqrt{2}$
2. $\frac{4\sqrt{3}}{3}$
3. $4\sqrt{3}$
4. $4$

Q. Two players A and B throw a die alternately for a prize of Rs 11, which is to be won by a player who first throws a six. If A starts the game, their respective expectations are

1. Rs 6; Rs 5
2. Rs 7; Rs 4
3. Rs 5.50; Rs 5.50
4. Rs 5.75; Rs 5.25

Q. Let $B_i(i=1,2,3)$ be three independent events in a sample space. The probability that only $B_1$ occur is $\alpha$, only $B_2$ occurs is $\beta$ and only $B_3$ occurs is $\gamma$. Let $p$ be the probability that none of the events $B_i$ occurs and these 4 probabilities satisfy the equations $(\alpha -2\beta )p=\alpha \beta$ and $(\beta -3\gamma )p=2\beta \gamma$ (All the probabilities are assumed to lie in the interval $(0,1))$. Then $\frac{P\left(B_1\right)}{P\left(B_3\right)}$ is equal to