Q82 / 160 · TS EAMCET 2021 (04 Aug Shift 1) · Mathematics

A fair coin is tossed $15$ times. The probability that the tail will appear atleast thrice is

Solution

The probability of getting

k

successes is

P (r \geq k) = \sum_{r = k}^{n} {}^{n}{C_{r} p^{r} q^{n - r}}

Where $n$ is number of trials, $p & q$ are probability of success and failure respectively.

Here, $n = 15, p = \frac{1}{2}, q = \frac{1}{2}$

Therefore,

$P (atleast 3 tails) = P (3 \geq r)$

$P = 1 - P (r = 0) - P (r = 1) - P (r = 2)$

$P = 1 - {}^{15}{C_{0} p^{0} q^{15} -} {}^{15}{C_{1} p^{1} q^{14} -} {}^{15}{C_{2} p^{2} q^{13}}$

$P = 1 - {(\frac{1}{2})}^{15} - 15 {(\frac{1}{2})}^{15} - 105 {(\frac{1}{2})}^{15}$

$P = 1 - \frac{121}{2^{15}}$

Hence, the correct answer is option: 'B'.