Questions: A person tosses a coin 16 times. In how many ways can he get 15 tails?

Solution Steps

We are tasked with finding the number of ways a person can obtain 15 tails when tossing a coin 16 times. This scenario can be modeled using a binomial distribution where:

• \( n = 16 \) (the number of trials),
• \( x = 15 \) (the number of successes, which in this case is getting tails),
• \( p = 0.5 \) (the probability of getting tails in a single toss),
• \( q = 0.5 \) (the probability of getting heads).

The probability of getting exactly \( x \) successes in \( n \) trials is given by the formula: \[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \] Substituting the values, we find: \[ P(X = 15) = \binom{16}{15} \cdot (0.5)^{15} \cdot (0.5)^{1} = \binom{16}{15} \cdot (0.5)^{16} \] Calculating this gives: \[ P(X = 15) = 0.0002 \]

The number of ways to achieve exactly 15 tails in 16 tosses is represented by the binomial coefficient \( \binom{n}{x} \): \[ \binom{n}{x} = \frac{n!}{x!(n-x)!} \] For our case: \[ \binom{16}{15} = \frac{16!}{15! \cdot 1!} = 16 \] Thus, the number of ways to get 15 tails in 16 tosses is: \[ \text{Number of ways} = 16 \]

Final Answer