Driving to work alone. It is reported that 77% of workers aged 16 and over drive to work alone. Choose 8 workers at random. Find the probability that
a) All drive to work alone
b) More than one-half drive to work alone
c) Exactly 3 drive to work alon
a).1235
b).9120
c).0165
To calculate the probability in these scenarios, we'll need to use the concept of binomial probability. First, let's define the variables:
n = total number of workers chosen (8)
p = probability of an individual worker driving alone (77% or 0.77)
a) Probability that all 8 workers drive to work alone:
To find the probability that all workers drive to work alone, we need to find the probability for each worker and then multiply them together since these events are independent.
P(all drive alone) = P(drive alone) * P(drive alone) * ... * P(drive alone)
P(all drive alone) = (0.77)^8
You can calculate this by raising the probability (0.77) to the power of the number of workers (8).
b) Probability that more than one-half drive alone:
To find the probability that more than half the workers drive alone, we need to calculate the probability of each possible scenario where more than four workers drive alone and add them together.
P(more than one-half drive alone) = P(5 drive alone) + P(6 drive alone) + P(7 drive alone) + P(8 drive alone)
P(n drive alone) = (nCr) * (p^n) * ((1-p)^(8-n))
Here, nCr represents the combination function for selecting n items out of 8 items.
c) Probability that exactly 3 workers drive alone:
To find the probability that exactly 3 workers drive alone, we once again need to calculate the probability of this specific scenario.
P(exactly 3 drive alone) = P(3 drive alone) = (8C3) * (0.77^3) * ((1-0.77)^(8-3))
Here, 8C3 represents the combination function for selecting 3 items out of 8 items.
Using these formulas and calculating the respective probabilities will provide you with the answers to each part of the question.
To find the probability in each case, we need to use the binomial probability formula:
P(X=k) = (n C k) * p^k * (1-p)^(n-k)
where:
- P(X=k) is the probability of k successes,
- n is the total number of trials (in this case, the number of workers selected),
- k is the number of desired successes,
- p is the probability of a single success (in this case, the probability that a worker drives to work alone),
- (n C k) represents the number of combinations of n items taken k at a time.
Given that 77% of workers drive to work alone, we can calculate p as 0.77.
a) All drive to work alone:
- k = 8 (since we want all 8 workers to drive alone)
- n = 8 (since we are selecting 8 workers)
- p = 0.77
P(X=8) = (8 C 8) * 0.77^8 * (1-0.77)^(8-8)
Calculating this:
P(X=8) = (1) * 0.77^8 * (1-0.77)^(0)
= 0.77^8
≈ 0.206
Therefore, the probability that all 8 workers drive to work alone is approximately 0.206.
b) More than one-half drive to work alone:
To find this probability, we need to sum up the probabilities of having more than half of the workers driving to work alone.
P(X > 4) = P(X=5) + P(X=6) + P(X=7) + P(X=8)
P(X>4) = (8 C 5) * 0.77^5 * (1-0.77)^(8-5)
+ (8 C 6) * 0.77^6 * (1-0.77)^(8-6)
+ (8 C 7) * 0.77^7 * (1-0.77)^(8-7)
+ (8 C 8) * 0.77^8 * (1-0.77)^(8-8)
Calculating this:
P(X>4) ≈ 0.522
Therefore, the probability that more than one-half of the workers drive to work alone is approximately 0.522.
c) Exactly 3 drive to work alone:
- k = 3
- n = 8
- p = 0.77
P(X=3) = (8 C 3) * 0.77^3 * (1-0.77)^(8-3)
Calculating this:
P(X=3) ≈ 0.215
Therefore, the probability that exactly 3 workers drive to work alone is approximately 0.215.