Question

Why is the probability of tossing a coin 10 times and getting 8 heads the same probability as tossing ten times and getting two heads

Answer 1

It's a neat property of binomial probabilities.

Answer 2

it helps that P(H)=P(T)

Answer 3

Yup

Answer 4

P(8H)=P(8T)=P(2H)

Answer 5

In terms of the binomial formula:
\[p_1=P(\text{8 heads in 10 tosses})=\binom{10}8\left(\frac{1}{2}\right)^8\left(\frac{1}{2}\right)^{2}\]
Knowing that \(\dbinom{10}8=\dbinom{10}2\), you have
\[p_2=P(\text{2 tails in 10 tosses})=\binom{10}2\left(\frac{1}{2}\right)^2\left(\frac{1}{2}\right)^{8}=p_1\]

Answer 6

because the number of ways to include 2 out of 10 people is the same as the number of ways to exclude 8 out of 10