What is the probability of getting exactly four heads when tossing a fair coin ten times?

To determine the probability of getting exactly four heads when tossing a fair coin ten times, we use the binomial probability formula, which is expressed as:

[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} ]

where:

• ( n ) is the number of trials (in this case, 10 coin tosses),

• ( k ) is the number of successful outcomes we are interested in (four heads),

• ( p ) is the probability of success on an individual trial (for a fair coin, ( p = 0.5 )),

• ( \binom{n}{k} ) represents the binomial coefficient, calculated as ( \frac{n!}{k!(n-k)!} ).

1. Calculate the binomial coefficient (\binom{10}{4}):

[

\binom{10}{4} = \frac{10!}{4!(10-4)!} = \frac{10!}{4!6!} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times