Question

A fair coin is tossed 99 times. If X is the number of times heads occurs P(X=r) is maximum when r, is
(This question has multiple correct options)
a) 49
b) 50
c) 51
d) None of these

Answer 1

Hint: A binomial distribution can be thought of as simply the probability of a SUCCESS or FAILURE outcome in an experiment or survey that is repeated multiple times. The binomial is a type of distribution that has two possible outcomes.

Complete step-by-step answer:

Let n be the total number of times a coin is tossed, p be the probability of the head of a single coin and q be the probability of the tail of a single coin.

Since X has binomial distribution with n = 99 and p = q = 0.5. We obtain that $P\left( X=r \right)={}^{n}{{C}_{r}}{{p}^{r}}{{q}^{n-r}}$ , where ${}^{n}{{C}_{r}}=\dfrac{n!}{r!(n-r)!}$
 $P\left( X=r \right)={}^{n}{{C}_{r}}{{(0.5)}^{r}}{{(0.5)}^{n-r}}={}^{n}{{C}_{r}}{{(0.5)}^{n-r+r}}={}^{n}{{C}_{r}}{{(0.5)}^{n}}={{(0.5)}^{n}}{}^{n}{{C}_{r}}$

Thus maximum of P(X = r) corresponds to the value of r with maximal

${}^{n}{{C}_{r}}=\dfrac{n!}{r!(n-r)!}$

Note that binomial coefficients are symmetric, ${}^{n-r}{{C}_{r}}={}^{n}{{C}_{r}}=\dfrac{n!}{r!(n-r)!}$

When the maximum corresponds to the points most closely to $\dfrac{n}{2}$ .

Since n is odd, there are two such values

${{r}_{1}}=\dfrac{n-1}{2}=\dfrac{98}{2}=49$

  and ${{r}_{2}}=\dfrac{n+1}{2}=\dfrac{100}{2}=50$

For these numbers the probability function P(X = r) has maximum.

Therefore, the correct options for the given question are option (a) and option (b).

Note: Criteria for a Binomial Probability distribution. A fixed number of trials and each trial are independent of the others. There are only two outcomes. The probability of each outcome remains constant from trial to trial.