a survey from teeneage research unlimited found that 30% of teenage consumers receive their spending money from parttime jobs. If 5 teenagers are selected at random, find the probability that at least 3 of them have part time jobs
To study
To find the probability that at least 3 of the 5 randomly selected teenagers have part-time jobs, we need to calculate the probability of three, four, or five teenagers having part-time jobs.
Step 1: Calculate the probability of exactly 3 teenagers having part-time jobs.
The probability of 3 teenagers having part-time jobs is given by:
(0.30)^3 * (0.70)^2 * C(5, 3)
where "^" denotes exponentiation, "C(n, r)" denotes the binomial coefficient (also known as combinations), which calculates the number of ways to choose r items from a set of n items, and (0.30)^3 represents the probability of a teenager having a part-time job, raised to the power of 3 (since we want exactly 3 of them to have part-time jobs). (0.70)^2 represents the probability of a teenager not having a part-time job, raised to the power of 2 (since we want the remaining 2 teenagers to not have part-time jobs). The binomial coefficient C(5, 3) calculates the number of ways to choose 3 teenagers from a set of 5 teenagers.
Step 2: Calculate the probability of exactly 4 teenagers having part-time jobs.
The probability of 4 teenagers having part-time jobs is given by:
(0.30)^4 * (0.70)^1 * C(5, 4)
Similarly, we raise (0.30)^4 to the power of 4 (since we want exactly 4 of them to have part-time jobs) and (0.70)^1 to the power of 1 (since we want the remaining 1 teenager to not have a part-time job). The binomial coefficient C(5, 4) calculates the number of ways to choose 4 teenagers from a set of 5 teenagers.
Step 3: Calculate the probability of exactly 5 teenagers having part-time jobs.
The probability of 5 teenagers having part-time jobs is given by:
(0.30)^5 * (0.70)^0 * C(5, 5)
Once again, we raise (0.30)^5 to the power of 5 (since we want all 5 of them to have part-time jobs), and (0.70)^0 to the power of 0 (since we want none of the remaining teenagers to not have a part-time job). The binomial coefficient C(5, 5) calculates the number of ways to choose 5 teenagers from a set of 5 teenagers.
Step 4: Add up the probabilities from Steps 1, 2, and 3 to get the final probability.
The final probability is the sum of probabilities calculated in Steps 1, 2, and 3:
P(at least 3 teenagers have part-time jobs) = P(3 having jobs) + P(4 having jobs) + P(5 having jobs)
So, you will calculate each of the three probabilities using the given formulas and then add them up to get the final probability.