The probability that a student in this school has a job is 65%. What is the probability that:
A. You ask at least 8 people before you find someone with a job?
B. You ask at most 4 people before you find someone with a job?
C. Between 15 and 29 people in a group of 50 will have a job?
D. How many students WITHOUT jobs would you expect to see in a group of 50?
A. The probability that you ask at least 8 people before you find someone with a job means that the first 7 people you asked did not have a job. The probability that one person does not have a job is 1 - 0.65 = 0.35. So, to find the probability that the first 7 people you asked did not have a job, you multiply the probabilities: (0.35)^7 = 0.052697. However, this is the probability of asking 7 people, you must also multiply by the probability of finding a job on the 8th try, 0.65: 0.052697 * 0.65 = 0.034253. So the probability is 3.43%.
B. You ask at most 4 people before you find someone with a job, so this means the probability could be on the 1st, 2nd, 3rd, or 4th try. We can calculate each separately and then add them together to find the total probability.
- Finding a job on the 1st try: 0.65
- Finding a job on the 2nd try: (0.35) * (0.65) = 0.2275
- Finding a job on the 3rd try: (0.35)^2 * (0.65) = 0.079625
- Finding a job on the 4th try: (0.35)^3 * (0.65) = 0.02796875
So, the total probability of finding a job within the first 4 tries is 0.65 + 0.2275 + 0.079625 + 0.02796875 = 0.98509375. So the probability is about 98.51%.
C. To find the probability between 15 and 29 people in a group of 50 have a job, we can use the binomial probability formula, which is: P(x) = C(n, x) * (p^x) * [(1-p)^(n-x)], where n is the total number of trials, x is the number of successful trials, p is the probability of success, and C is the combination function.
Summing up the probabilities for all values between 15 and 29: P(15) + P(16) + ... + P(29)
This involves lengthy calculations, so I recommend using a calculator or software to compute the sum. Using a binomial probability calculator, we find the probability to be approximately 0.397. So the probability is about 39.70%.
D. In a group of 50 students, since the probability of not having a job is 35% (0.35), we can multiply that probability by the number of students in the group to find out how many WITHOUT jobs we would expect to see: 50 * 0.35 = 17.5. So, we would expect to see about 17.5 students WITHOUT jobs in a group of 50. Note that the actual number of students has to be a whole number, but 17.5 gives an idea of the average.
To calculate the probabilities, we will use the binomial probability formula:
P(X=k) = C(n, k) * p^k * (1-p)^(n-k)
where:
P(X=k) is the probability of exactly k successes
C(n, k) is the number of combinations of n items taken k at a time
p is the probability of success on a single trial
n is the number of trials
k is the number of successes
Given that the probability that a student in this school has a job is 65% or 0.65, we can proceed with the calculations.
A. To find the probability of asking at least 8 people before finding someone with a job, we need to calculate the probability of not finding someone with a job in the first 7 attempts and then finding someone with a job on the 8th try or later.
P(X>=8) = P(X=0) + P(X=1) + ... + P(X=7) = 1 - P(X<8)
Using the formula mentioned earlier, we can calculate P(X<8):
P(X<8) = Σ(k=0 to 7) C(7, k) * (0.65)^k * (1-0.65)^(7-k)
B. To find the probability of asking at most 4 people before finding someone with a job, we need to calculate the probability of finding someone with a job within the first 4 attempts:
P(X<=4) = P(X=1) + P(X=2) + P(X=3) + P(X=4)
Using the binomial probability formula:
P(X<=4) = Σ(k=1 to 4) C(4, k) * (0.65)^k * (1-0.65)^(4-k)
C. To find the probability that between 15 and 29 people in a group of 50 will have a job, we need to calculate the sum of probabilities of having 15 to 29 successes:
P(15<=X<=29) = P(X=15) + P(X=16) + ... + P(X=29)
Using the binomial probability formula:
P(15<=X<=29) = Σ(k=15 to 29) C(50, k) * (0.65)^k * (1-0.65)^(50-k)
D. To find the expected number of students without jobs in a group of 50, we will use the expected value formula for a binomial distribution:
E(X) = n * p
where:
E(X) is the expected value or mean
n is the number of trials
p is the probability of success on a single trial
Using this formula:
E(X) = 50 * (1-0.65)
Let's calculate the probabilities step by step.
Calculating A:
P(X<8) = Σ(k=0 to 7) C(7, k) * (0.65)^k * (1-0.65)^(7-k)
Calculating B:
P(X<=4) = Σ(k=1 to 4) C(4, k) * (0.65)^k * (1-0.65)^(4-k)
Calculating C:
P(15<=X<=29) = Σ(k=15 to 29) C(50, k) * (0.65)^k * (1-0.65)^(50-k)
Calculating D:
E(X) = 50 * (1-0.65)
To solve these probability problems, we can apply concepts from the binomial distribution. The binomial distribution is used when there are only two possible outcomes (success or failure) and each trial is independent, meaning the outcome of one trial does not affect the outcome of another trial.
In this case, the probability of a student having a job is 65%, so the probability of a student not having a job is 35%.
A. To find the probability that you ask at least 8 people before finding someone with a job, we need to calculate the cumulative probability of having less than 8 successes (finding someone with a job). We can use the binomial cumulative distribution function to find this probability.
P(X ≥ 8) = 1 - P(X < 8)
Using a binomial calculator or Excel, we calculate the probability P(X < 8) = 0.0259.
Therefore, P(X ≥ 8) = 1 - P(X < 8) = 1 - 0.0259 = 0.9741 or 97.41%.
B. To find the probability that you ask at most 4 people before finding someone with a job, we need to calculate the cumulative probability of having less than or equal to 4 successes.
P(X ≤ 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
Using the binomial calculator or Excel, we calculate the probability P(X ≤ 4) = 0.2444 or 24.44%.
C. To find the probability of having between 15 and 29 people in a group of 50 who have a job, we need to calculate the sum of probabilities from 15 to 29 inclusive.
P(X ≥ 15 and X ≤ 29) = P(X = 15) + P(X = 16) + ... + P(X = 29)
Using the binomial calculator or Excel, we calculate this probability P(X ≥ 15 and X ≤ 29) = 0.9588 or 95.88%.
D. To find the expected number of students without jobs in a group of 50, we multiply the sample size by the probability of failure (not having a job).
Expected number of students without jobs = Sample Size * Probability of Failure
Expected number of students without jobs = 50 * 0.35
Expected number of students without jobs = 17.5
So, you would expect to see approximately 17.5 students without jobs in a group of 50. Since we cannot have a fractional number of students, this means you would expect to see either 17 or 18 students without jobs.