Find each probability if three coins are tossed:
- P($3$ heads or $3$ tails)
- P(exactly $2$ tails)
When a coin is tossed,
probability of getting head $= p = \dfrac{1}{2}$
probability of getting tail $= q = 1- p = \dfrac{1}{2}$
Number of coins $= 3$
- P($3$ heads or $3$ tails)
$= P \left(3 \text{ heads}\right) + P \left(3 \text{ tails}\right)$
$= {^{3}}{C}_{3} \times \left(\dfrac{1}{2}\right)^3 \times \left(\dfrac{1}{2}\right)^0 + {^{3}}{C}_{0} \times \left(\dfrac{1}{2}\right)^0 \times \left(\dfrac{1}{2}\right)^3$
$= 2 \times \left(\dfrac{1}{2}\right)^3$
$= \dfrac{1}{4}$
- P(exactly $2$ tails)
$= P \left(\text{exactly }1 \text{ head}\right)$
$= {^{3}}{C}_{1} \times \left(\dfrac{1}{2}\right)^1 \times \left(\dfrac{1}{2}\right)^2$
$= 3 \times \dfrac{1}{8} = \dfrac{3}{8}$