According to a previous report, 25% of households had some type of high-speed Internet connection. Suppose 8 households are selected at random and the number of households with high-speed Internet is recorded (a) Find the probability that exactly 3 households have high-speed Internet (b) Find the probability that at least 5 households have high-speed Internet. (c) Find the probability that fewer than 3 households have high-speed Internet. (d) Find the probability that between 2 and 5 households, inclusive, have high-speed Internet.
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To solve this problem, we will use the binomial probability formula. The binomial probability is used when there are two possible outcomes (success or failure) for each trial, and the trials are independent and repeated a fixed number of times.
In this case, the success is having a high-speed internet connection, and the failure is not having a high-speed internet connection. The probability of success is given as 25%, which is equivalent to 0.25. The number of trials is 8 households.
(a) To find the probability that exactly 3 households have a high-speed internet connection, we need to calculate the binomial probability for exactly 3 successes in 8 trials.
The formula for binomial probability is:
P(X = k) = (n Choose k) * p^k * (1 - p)^(n - k)
Where:
- (n Choose k) is the number of combinations of n items taken k at a time, and can be calculated as:
(n Choose k) = n! / (k!(n - k)!)
- p is the probability of success in one trial
- n is the number of trials
Plugging in the values:
P(X = 3) = (8 Choose 3) * 0.25^3 * (1 - 0.25)^(8 - 3)
Using the binomial coefficient calculation and simplifying the equation, we get:
P(X = 3) = 56 * 0.015625 * 0.2373046875
Calculating the expression, we find:
P(X = 3) ≈ 0.2637
Therefore, the probability that exactly 3 households have a high-speed internet connection is approximately 0.2637.
(b) To find the probability that at least 5 households have a high-speed internet connection, we need to sum up the probabilities for 5, 6, 7, and 8 successes.
P(X ≥ 5) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8)
Using the binomial probability formula as described above, calculate the probabilities for each case, and sum them up.
(c) To find the probability that fewer than 3 households have a high-speed internet connection, we need to calculate the probabilities for 0, 1, and 2 successes and sum them up.
P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)
Using the binomial probability formula, calculate the probabilities for each case, and sum them up.
(d) To find the probability that between 2 and 5 households (inclusive) have a high-speed internet connection, we need to calculate the probabilities for 2, 3, 4, and 5 successes and sum them up.
P(2 ≤ X ≤ 5) = P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
Using the binomial probability formula, calculate the probabilities for each case, and sum them up.