Binomial Probability Theorem

Q.

A coin is tossed $3$ times by $2$ persons. What is the probability that both get an equal number of heads

1. $\frac{3}{8}$

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

3. $\frac{5}{16}$

4. None of these

Q.

The sum of two positive numbers is $100$. The probability that their product is greater than $1000$ is

1. $\frac{7}{9}$

2. $\frac{7}{10}$

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

4. None of the above.

Q. A box has 50 pens of which 20 are defective. What is the probability that out of a sample of 5 pens drawn one by one with replacement, at most one is defective?

1. $3.6\times {\left(\frac{3}{5}\right)}^4$
2. ${\left(\frac{3}{5}\right)}^5$
3. $2.6\times {\left(\frac{3}{5}\right)}^4$
4. ${\left(\frac{2}{5}\right)}^5$

Q.

A man takes a step forward with probability 0.4 and backward with probability 0.6. Find the probability that at the end of eleven steps, he is one step away from the starting point.

1. $=462\times (0.4{)}^6\times (0.6{)}^4$

2. $=462\times (0.24{)}^{10}$

3. $=462\times (0.4{)}^4\times (0.6{)}^6$

4. $=462\times (0.24{)}^5$

Q. 31. One dice is thrown three times and the sum of the thrown numbers is 15. The probability for which number 4 appears in the first throw is 1/18. Prove it.

Q. The probability of occurrence of an event $A$ in one trial is $0.4$. The probability that the event $A$ happens at least once in three independent trials is:

1. $1-0.784$
2. $0.784$
3. $1-0.216$
4. $0.216$

Q. A multiple choice examination has $5$ questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get $4$ or more correct answers just by guessing is :

1. $\frac{17}{3^5}$
2. $\frac{13}{3^5}$
3. $\frac{11}{3^5}$
4. $\frac{10}{3^5}$

Q. Minimum number of times a fair coin must be tossed so that the probability of getting at least one head is more than $99\%$ is :

1. $5$
2. $6$
3. $7$
4. $8$

Q. A biased coin is thrown 100 times with probability of head = p and probability of tail = q. The probability of getting 55 heads in 100 trials is $\frac{100!}{55!\times 45!}{p}^{55}{q}^{45}$

1. True
2. False

Q. Match the following by appropiately matching the lists based on the information given in Column I and Column II.
​​​​​​$\begin{array}{cc}Column1& Column2\\ \text{(a) The probability of a bomb hitting a bridge is}& \\ \frac{1}{2}.\text{Two direct hits are needed to destroy it.}& \\ \text{The number of bombs recquired so that the}& \\ \text{probability of the bridge being destroyed is}& \\ \text{greater than}0.9\text{can be}& \text{(p)}4\\ \text{(b) A bag contains}2\text{red,}3\text{white,}5\text{black balls, a}& \\ \text{ball is drawn its color is noted and replaced}& \\ \text{The number of times, a ball can be drawn}& \\ \text{so that the probability of getting a red ball}& \\ \text{for the first time is atleast}1/2& \text{(q)}6\\ \text{(c) A drawer contains a mixture of red socks}& \\ \text{and blue socks, at most}17\text{in all. It so}& \\ \text{happens that when two socks are selected}& \\ \text{randomly without replacement, there is a}& \\ \text{probability of exactly}1/2\text{that both are red}& \\ \text{or both are blue. Then number of red}& \\ \text{socks in drawer can be}& \text{(r)}7\\ \text{(d) There are two red, two blue, two white and}& \\ \text{certain number (greater than}0\text{) of green}& \\ \text{socks in a drawer. If two socks are taken at}& \\ \text{random from the drawer without}& \\ \text{replacement, the probability that they}& \\ \text{are of same color is}\frac{1}{5}\text{, then the number of}& \\ \text{green socks are}& \text{(s)}10\end{array}$

1. $a\to r,s;b\to p,q,r,s;c\to p,q,r,s;d\to p$
2. $a\to r,s;b\to p,q,s;c\to p,q,r,s;d\to p$
3. $a\to r,s;b\to p,s;c\to p,s;d\to p$
4. $a\to r,s;b\to p,q,r,s;c\to p,q,r,s;d\to q,r,s$

Q. In a exam there are $30$ true/false questions. If a student guesses all the $30$ questions, then the probability that he/she gets atleast $15$ correct, is

1. $\frac{1}{2}$
2. $1-\frac{{}^{30}{C}_{15}}{2^{30}}$
3. $\frac{1}{2}+\frac{{}^{30}{C}_{15}}{2^{31}}$
4. $\frac{1}{2}+\frac{{}^{30}{C}_{15}}{2^{30}}$

Q. A box contains 100 bulbs, 20 of which are defective. 10 bulbs are selected for inspection. Find the probability that:
(i) all 10 are defective
(ii) all 10 are good
(iii) at least one is defective
(iv) none is defective

Q. A box contains $12$ white and $12$ black balls. The balls are drawn at random from the box, one at a time with replacement. The probability that a white ball is drawn for the fourth time on the seventh draw, is

1. $\frac{5}{64}$
2. $\frac{27}{32}$
3. $\frac{5}{32}$
4. $\frac{1}{2}$

Q. A and B play a game where each is asked to select a number from 1 to 25. If the numbers selected by A and B match, both of them win a prize. The probability that they win their third prize on $5^{th}$ game is equal to:

1. $\frac{6.(24{)}^2}{(25{)}^5}$
2. $\frac{6.(21{)}^2}{(25{)}^5}$
3. $\frac{(24{)}^2}{(25{)}^2}$
4. $\frac{6.(24{)}^2}{(25{)}^4}$

Q. A fair coin is tossed $100$ times. The probability of getting tails $1,3,....49$ times is

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

Q. One hundred identical coins, each with probability p of showing up heads are tossed once. If 0 < p < 1 and the probability of heads showing 50 coins is equal to that head showing 51 coins, then the value of p is

1. $\frac{1}{2}$
2. $\frac{49}{101}$
3. $\frac{50}{101}$
4. $\frac{51}{101}$

Q. An ordinary cube has 4 blank faces, one face marked 2 and another marked 3. Then the probability of obtaining 12 in 5 throws is:

1. $\frac{5}{1296}$
2. $\frac{5}{1944}$
3. $\frac{5}{2592}$
4. $None$

Q. If a year of 22nd century is randomly selected, find out the probability of the year having 53 sundays.

Q. A can hit a target 4 times on 5 shots B can hit 3 times on 4 shots and C can hit 2 on 3 shots , wats the probability tat all may hit

Q. A box contains 10 good articles and 6 defective articles. One item is drawn at random. The probability that it is either good or has a defect, is
(a) 64/64
(b) 49/64
(c) 40/64
(d) 24/64

Q. An experiment consists of three throws of coin and success means two heads. The probability of no success, if experiment is repeated three times, is

1. $\frac{1}{25}$
2. $\frac{2}{5}$
3. $\frac{24}{25}$
4. None

Q. There are two die $A$ and $B$ both having six faces. Die $A$ has three faces marked with $1$, two faces marked with $2$, and one face marked with $3$. Die $B$ has one face marked with $1$, two faces marked with $2$, and three faces marked with $3$. Both dices are thrown randomly once. If $E$ be the event of getting sum of the numbers appearing on top faces equal to $x$, let $P\left(E\right)$ be the probability of event $E$, then
$P\left(E\right)$ is maximum when $x$ equal to

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

Q. From a well-shuffled deck of $52$ cards, $9$ cards are taken out one by one with replacement. Then the probability that out of $9$ cards, $5$ cards are spades, is

1. $126\cdot \frac{81}{4^9}$
2. $126\cdot \frac{27}{4^9}$
3. $84\cdot \frac{81}{4^9}$
4. $84\cdot \frac{27}{4^9}$

Q. A box contains 30 apples and 40 oranges half of apple and half of orange are rotten if two items are drawn at random what is the probability of either both are rotten or both are apples

Q. There are two die $A$ and $B$ both having six faces. Die $A$ has three faces marked with $1$, two faces marked with $2$, and one face marked with $3$. Die $B$ has one face marked with $1$, two faces marked with $2$, and three faces marked with $3$. Both dices are thrown randomly once. If $E$ be the event of getting sum of the numbers appearing on top faces equal to $x$, let $P\left(E\right)$ be the probability of event $E$, then
When $x=4$, then $P\left(E\right)$ is equal to

1. $\frac{5}{9}$
2. $\frac{6}{7}$
3. $\frac{7}{18}$
4. $\frac{8}{19}$

Q. Three of the six vertices of a regular hexagon are chosen at random. What is the probability that the triangle with these vertices is equilateral.

Q. A contest consists of predicting the results win, draw or defeat of 7 foot ball matches. A sent his entry by predicting at random. The probability that his entry will contain exactly 4 correct predictions is:

1. $\frac{8}{3^7}$
2. $\frac{16}{3^7}$
3. $\frac{280}{3^7}$
4. $\frac{560}{3^7}$