Question

An unbiased coin is tossed 15 times. in how many ways can the coin land tails either exactly 5 times or exactly 3 times?

Answer 1

THis is a binomial distribution:-

P( 3 tails) = 15C3 (0.5)^3(0.5)^12

=  0.0139

P(5 tails) = 15C5 (0.5)^5(0.5)^10

= 0.0916

P(3 or 5) = 0.0139 + 0.0916 = 0.1055  Answer

Answer 2

Number of times = 15

$\text {Probability of tails exactly 5 times :}$
$\left(\begin{array}{cc}15\\3\end{array}\right) \bigg( \dfrac{1}{2} \bigg)^{15} = \dfrac{3003}{32768}$

$\text {Probability of tails exactly 3 times :}$
$\left(\begin{array}{cc}15\\3\end{array}\right) \bigg( \dfrac{1}{2} \bigg)^{15} = \dfrac{455}{32768}$

Total probability:
$\dfrac{3003}{32768} + \dfrac{455}{32768} = \dfrac{1729}{16384} = 0.11 \text { \sqrt{x} (nearest hundred)}$