A survey shows that 25% of people listen to the radio daily. If a group of 125 people are surveyed, what is the probability that more than 40 people will have listened to the radio today?
To find the probability that more than 40 people will have listened to the radio, we first need to determine the probability that 40 or fewer people will have listened to the radio.
Given that 25% of people listen to the radio daily, the probability that a randomly selected person listens to the radio on any given day is 0.25.
The number of people who listened to the radio in a group of 125 people follows a binomial distribution with parameters n = 125 (number of trials) and p = 0.25 (probability of success).
To find the cumulative probability of 40 or fewer people listening to the radio, we sum up the individual probabilities of having 0, 1, 2,..., 40 people listening to the radio.
The probability mass function of a binomial distribution is given by the formula:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- X is the random variable denoting the number of people listening to the radio
- k is the number of people listening to the radio (where k can range from 0 to 40)
- C(n, k) is the binomial coefficient (combining n and k)
We can use this formula to calculate the probability of 40 or fewer people listening to the radio:
P(X ≤ 40) = P(X = 0) + P(X = 1) + P(X = 2) + ... + P(X = 40)
We can substitute the values into the binomial distribution formula and calculate the probabilities for each value of k. Summing up these probabilities will give us P(X ≤ 40).
P(X ≤ 40) = Σ [C(125, k) * 0.25^k * (1 - 0.25)^(125 - k)] for k = 0 to 40
After calculating P(X ≤ 40), we can find the probability of more than 40 people listening to the radio by subtracting P(X ≤ 40) from 1:
P(X > 40) = 1 - P(X ≤ 40)