Final answer:
To find the probability that Donna Prima will play tonight, we use a Poisson distribution with an average rate of eight coughs occurring just before her performance. By calculating the complement of P(X ≤ 5) using the Poisson probability mass function, we can find the probability that she will play.
Step-by-step explanation:
To find the probability that Donna Prima will play tonight, we need to calculate the probability of hearing more than five coughs. We know that on average, eight coughs occur just before her performance. We can use a Poisson distribution to calculate this probability.
A Poisson distribution is used to model the number of events occurring within a fixed interval of time or space. In this case, the fixed interval is just before Donna Prima's performance. The average rate of events is given as eight coughs.
The formula for the probability mass function of a Poisson distribution is: P(X = k) = (e^-λ * λ^k) / k!, where λ is the average rate of events and k is the number of events we're interested in.
In this case, we want to find P(X > 5), which is the probability of hearing more than five coughs. We can calculate this probability by finding the complement of P(X ≤ 5): P(X > 5) = 1 - P(X ≤ 5).
Substituting the given values, we get:
P(X > 5) = 1 - (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)).
Using the formula for the Poisson distribution, the probability mass function can be calculated for each value of k and then summed to find P(X ≤ 5). This can be done using a calculator or statistical software. Once P(X ≤ 5) is found, subtract it from 1 to find P(X > 5), which is the probability that Donna Prima will play tonight.