If 55% of a television viewing population watched a program called ¡§Name that
Poem¡¨ one evening, what is the probability that, in a random sample of 100
viewers, less than 50% of the sample watched the program?
To find the probability that less than 50% of the random sample watched the program, we need to use the concept of binomial probability.
The binomial probability formula is:
P(X = k) = (n choose k) * p^k * q^(n-k)
Where:
- P(X = k) is the probability of getting exactly k successes in n trials
- (n choose k) represents the number of ways to choose k successes from n trials
- p is the probability of success on each trial
- q is the probability of failure on each trial (1 - p)
In this case, we want to find the probability that less than 50% of the sample watched the program. So, we need to calculate the probability of getting 0, 1, 2, ..., 49 successes in a sample of 100 viewers and sum them up.
Let's break down the steps to find the probability:
1. Calculate the probability of success, p, which is given as 55% or 0.55.
2. Calculate the probability of failure, q, which is 1 - p or 1 - 0.55 = 0.45.
3. Determine the number of trials, n, which is 100.
4. Calculate the probability for each number of successes, k, from 0 to 49 using the formula mentioned above.
5. Sum up the probabilities calculated in step 4.
You can use a calculator or software like Excel or Google Sheets to calculate each term in the sum and find the final probability.
Note: The binomial probability assumes that the trials are independent and the probability of success is the same for each trial.