Probability Distribution

Q.

What is the standard deviation of the data given below?

$10,28,13,18,29,30,22,23,25,and32$

Q.

In a box of $10$ electric bulbs, two are defective. Two bulbs are selected at random one after the other from the box. The first bulb after selection being put back in the box before making the second selection. The probability that both the bulbs are without defect is

1. $\frac{9}{25}$

2. $\frac{16}{25}$

3. $\frac{4}{5}$

4. $\frac{8}{25}$

Q.

In a box containing $100$ eggs, $10$ eggs are rotten. The probability that out of a sample of $5$ eggs none is rotten if the sampling is with replacement is

1. ${\left(\frac{1}{10}\right)}^5$

2. ${\left(\frac{1}{5}\right)}^5$

3. ${\left(\frac{9}{5}\right)}^5$

4. ${\left(\frac{9}{10}\right)}^5$

Q. Find the mean number of heads in three tosses of a fair coin.

Q. An unbiased coin is tossed $5$ times. Suppose that a variable $X$ is assigned the value $k$ when $k$ consecutive heads are obtained for $k=3,4,5$, otherwise $X$ takes the value $-1$. The expected value of $X$, is

1. $\frac{1}{8}$
2. $\frac{3}{16}$
3. $-\frac{1}{8}$
4. $-\frac{3}{16}$

Q. In a game, a man wins Rs. $100$ if he gets $5$ or $6$ on a throw of a fair die and loses Rs. $50$ for getting any other number on the die. If he decides to throw the die either till he gets a five or a six or to a maximum of three throws, then his expected gain/loss (in rupees) is :

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

Q.

A die is tossed twice.

Getting a number greater than $4$ is considered a success.

Then the variance of the probability distribution of the number of successes is.

1. $\frac{2}{9}$

2. $\frac{4}{9}$

3. $\frac{1}{3}$

4. None of these

Q.

A box contains $10$ red balls and $15$ green balls. If two balls are drawn in succession then the probability that one is red and other is green is

1. $\frac{1}{3}$

2. $\frac{1}{2}$

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

4. None of these

Q.

A bag contains $4$ red and $6$ black balls. A ball is drawn at random from the bag, its colour is observed and this ball along with two additional balls of the same colour are returned to the bag. If now a ball is drawn at random from the bag, then the probability that this drawn ball is red, is

1. $\frac{1}{5}$

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

3. $\frac{3}{10}$

4. $\frac{2}{5}$

Q.

Find the probability distribution of

number of tails in the simultaneous tosses of three coins

Q. Let a biased coin be tossed $5$times. If the probability of getting $4\text{heads}$ is equal to the probability of getting $5\text{heads}$, then the probability of getting atmost two heads is:

Q.

$AandB$ play a game where each is asked to select a number from $1to25.$ If the two numbers match, both of them win a prize. The probability that they will not win a prize in a single trial is

1. $\frac{1}{25}$

2. $\frac{24}{25}$

3. $\frac{2}{25}$

4. None of these

Q. On a normal standard die, one of the $21$ dots from any one of the six faces is removed at random with each dot equally likely to be chosen. The die is then rolled. If the probability that the top face has an odd number of dots is $\frac{p}{q}$ where $p$ and $q$ are in their lowest form, then the value of $\frac{(p+q)}{4}$ is

Q.

If a dice is thrown twice, the probability of occurrence of $4$ at least once is

1. $\frac{11}{36}$

2. $\frac{7}{12}$

3. $\frac{35}{36}$

4. None of these

Q. An unbalanced dice (with six faces numbered from $1$ to $6$) is thrown. The probability that the face value is odd is $90$% of the probability that the face value is even. The probability of getting any even numbered face is the same. If the probability that the face is even given that it is greater than $3$ is $0.75$, then the probability that the face value exceeds $3$ is

1. $\frac{80}{171}$
2. $\frac{29}{57}$
3. $\frac{82}{171}$
4. $\frac{9}{19}$

Q.

In a dice game, a player pays a stake of Rs 1 for each throw of a die. She receives Rs 5, if the die shows a 3, Rs 2, if the die shows a 1 or 6 and nothing otherwise, then what is the player's expected profit per throw over a long series of throws ?

Q.

Let $U_1$ and $U_2$be two urns such that $U_1$ contains $3$ white and $2$ red balls and $U_2$ contains only $1$ white ball. A fair coin is tossed. If the head appears then $1$ ball is drawn at random from $U_1$and put into $U_2$. However, if the tail appears then $2$ balls are drawn at random from $U_1$ and put into $U_2$. Now, $1$ ball is drawn at random from $U_2$. The probability of the drawn ball from $U_2$ being white is

1. $\frac{13}{30}$

2. $\frac{23}{30}$

3. $\frac{19}{30}$

4. $\frac{11}{30}$

Q. Eight coins are tossed together. The probability of getting exactly $3$ heads is

Q. Let $X$ have a binomial distribution $B(n,p)$ such that the sum and the product of the mean and variance of $X$ are $24\text{and}128$ respectively. If $P(X>n-3)=\frac{k}{2^n}$ , then $k$ is equal to :

Q.

A couple has two children.

Find the probability that both children are males, if it is known that atleast one of the children is male.

Find the probability that both children are females if it is known that the elder child is a female.

Q. The probability of an event happening in one trial of an experiment is 0.6. Three independent trial are made. The probability that the event happens at least once is:

1. 0.432
2. 0.064
3. 0.936
4. 0.568

Q.

The probabilities of two events $A$ and $B$ are given as $P\left(A\right)=0.8$ and $P\left(B\right)=0.7$. What is the minimum value of $P(A\cap B)$?

1. $0$

2. $0.1$

3. $0.5$

4. $1$

Q.

Assume $X$ is normally distributed with a mean of $12$ and a standard deviation of $2$.

Determine the value for $x$ that solves each of the following.

Round the answers to $2$ decimal places.

$\left(a\right)$ $\text{P}\left(X>x\right)=0.5$

$x=?$

$\left(b\right)$ $\text{P}\left(X>x\right)=0.95$

$x=?$

$\left(c\right)$ $\text{P}\left(x<X<12\right)=0.2$

$x=?$

$\left(d\right)$ $\text{P}\left(-x<X-12<x\right)=0.95$

$x=?$

$\left(e\right)$ $\text{P}\left(-x<X-12<x\right)=0.99$

$x=?$

Q. A random variable $X$ has the following probability distribution:
$$\begin{array}{cccccc}X& 1& 2& 3& 4& 5\\ P\left(X\right)& K^2& 2K& K& 2K& 5K^2\end{array}$$
Then $P(X>2)$ is equal to:

1. $\frac{7}{12}$
2. $\frac{23}{36}$
3. $\frac{1}{36}$
4. $\frac{1}{6}$

Q. The mean and standard deviation of random variable $X$ are $10$ and $5$ respectively. Then $E{\left(\frac{X-15}{5}\right)}^2=$______

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

Q. A random variable X has the following probability distribution. X 0 1 2 3 4 5 6 7 P (X) 0 k 2 k 2 k 3 k k 2 2 k 2 7 k 2 + k Determine (i) k (ii) P (X < 3) (iii) P (X > 6) (iv) P (0 < X < 3)

Q.

A biased die is tossed if an even face has turned up then the probability that it is face $2$ or face $4$ is

Face	$1$	$2$	$3$	$4$	$5$	$6$
Probability	$0.1$	$0.24$	$0.19$	$0.18$	$0.15$	$0.14$

1. $0.25$

2. $0.42$

3. $0.75$

4. $0.9$

Q. Find the probability distribution of number of doublets in three throws of a pair of dice.

Q. A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails.

Q.

How to solve Monty hall problem by using conditional probability?