Binomial Probability theorem

Q.

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

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

2. $\frac{51}{101}$

3. $\frac{49}{101}$

4. None of these

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

2. $=462\times (0.4{)}^6\times (0.6{)}^4$

3. $=462\times (0.24{)}^5$

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

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

Q. A coin whose faces marked 2 and 3 is thrown 5 times, then chance of obtaining a total of 12 is

1. $\frac{5}{16}$
2. $\frac{5}{8}$
3. $\frac{5}{32}$
4. $\frac{5}{24}$

Q. A fair coin is tossed n times and let X denotes the number of heads obtained. If P(X = 4), P(X = 5), and P(X = 6) are in A.P., then n is equal to

1. Only 7
2. 14
3. 7 or 14
4. 8

Q. Two players toss 4 coins each. The probability that they both obtain the same number of heads is

1. $\frac{5}{256}$
2. $\frac{1}{16}$
3. $\frac{35}{126}$
4. $\frac{3}{16}$

Q. A fair coin is tossed a fixed number of times. If the probability of getting 7 heads is equal to that getting 9 heads, then the probability of getting 3 heads is

1. $\frac{35}{2^{12}}$
2. $\frac{35}{2^{14}}$
3. $\frac{7}{2^{12}}$
4. $\frac{7}{2^{14}}$

Q. A pair of dice is thrown 5 times. If the probability getting a doublet twice is P(A), then the value $3888P\left(A\right)=$ ___

Q. Numbers are selected at random, one at a time, from the two-digit numbers 00, 01, 02, ..., 99 with replacement. An event E occurs if and only if the product of the two digits of a selected number is 18. If four numbers are selected, find probability that the event E occurs at least 3 times.

1. $\frac{96}{{25}^4}$
2. $\frac{1}{{25}^4}$
3. $\frac{97}{{25}^4}$
4. $\frac{95}{{25}^4}$

Q. The probability of success is p in one trial and the experiment be repeated n times. If the ratio of the probability of exactly r success to the probability of exactly r failures is independent of n and r, then p =

1. $\frac{1}{2}$
2. exists no such value of p
3. $\frac{1}{3}$
4. None of these

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

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 on 50 coins is equal to that of heads showing on 51 coins, then the value of p is:

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