The rates of on time flights for commercial jets are continuously tracked by the US Department of Transportation. Recently, Southwest Air had the best rate of 80% of its flight time on time. A test is conducted by randomly selecting 15 southwest flights and observing whether they arrive on time.

a. Find the probability that exactly 10 flights arrive on time.
b. Find probability that at least 10 flights arrive on time.
c. Find probability that at least 10 flights arrive late.
d. Would it be unusual for Southwest to have 5 flights arrive late? why or why not?

Find the probability of geting the outcome of head and a 5 when a coin is tossed and a single die is rolled.

An EXCEL spreadsheet is very helpful for these kinds of problems.

a) first calculate the number of ways 15 flights could be on time 10 times and late 5 times. The formula for for n-choose-x is n!/x!*(n-x)! where ! means factorial. So, 15-choose-5 becomes (11*12*13*14*15)/(1*2*3*4*5) = 3003. Multiply this times .8^10*.2^5 = .0000344. So the probability becomes .0000344*3003 = 10.32%

b) repeat methodology used in a for exactly 11, exactly 12, ... exactly 15.

c) repeat methodology in b for

Sorry i didn't finish.

Probability of a head on a coin and 5 on a die is (1/2) * (1/6)

Thanks!

Sorry but I don't really understand the first part.

Are you familiar with combinatorial problems of n-choose-x. If you have n items (in your case 15 flights) and you want to choose x of them (in your problem a. 10 flights) Counting, how many different ways can this be done. You would use the formula (n!)/(x!*(n-x)!) where ! means factorial. And n!=1*2*3*...n.

So, I calculate if there are 15 flights, and 5 of them are late (10 on time), there are 3003 different ways you could arrange this.

Lets take one of them. Say flights 1 to 10 are on time and flights 11 to 15 are late. What is the probability of seeing this pattern? It would be:
(0.80)^10 * (0.20)^5 = .0000344.
Still with me?
Lets take another. Say flight 1 is late, 2 to 11 are on time, and 12 to 15 are late. The probability of seeing this pattern is:
0.2 * (0.8)^10 * (0.2)^4 = .0000344

Since there are 3003 ways to arrange the late/on-time patterns and have 5 late and 15 on time, the overall probability of having exactly 5 late is .0000344*3003=.1032 = 10.32%

Using my EXCEL spreadsheet, I calculate the probabilities table (i hope my cut and paste works)

late possible probability
ways

0 1 0.035184372
1 15 0.131941395
2 105 0.230897442
3 455 0.250138895
4 1365 0.187604171
5 3003 0.103182294
6 5005 0.042992623
7 6435 0.013819057
8 6435 0.003454764
9 5005 0.00067176
10 3003 0.000100764
11 1365 1.14504E-05
12 455 9.54204E-07
13 105 5.50502E-08
14 15 1.96608E-09
15 1 3.2768E-11

This should give you all the info you need to answer the questions.

I hope this helps
lottsa luck

Clown Bot Clown Bot answered
9 months ago

a. The probability that exactly 10 flights arrive on time is 10.32%.

b. The probability that at least 10 flights arrive on time can be found by adding up the probabilities of having 10, 11, 12, 13, 14, or 15 flights arrive on time. This gives us a total probability of 0.000100764 + 1.14504E-05 + 9.54204E-07 + 5.50502E-08 + 1.96608E-09 + 3.2768E-11 = 0.000112147.

c. The probability that at least 10 flights arrive late is the complement of the probability of having at least 10 flights arrive on time. So it would be 1 - 0.000112147 = 0.999887853.

d. To determine if it would be unusual for Southwest to have 5 flights arrive late, we can compare the probability of this event to a predetermined threshold. If the probability is lower than the threshold, it would be considered unusual. However, since we do not have a threshold provided in the question, we cannot determine if it would be unusual or not.

Step-by-Step Bot Step-by-Step Bot answered
8 months ago

a. To find the probability that exactly 10 flights arrive on time, you need to calculate the probability of exactly 10 successes (flights arriving on time) out of 15 trials (selected flights). The formula to calculate this is 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 or ways to choose k successes out of n trials
p is the probability of success (flight arriving on time)
n is the number of trials (selected flights)

Using the formula, we can calculate it as follows:

P(X = 10) = C(15, 10) * (0.8)^10 * (1 - 0.8)^(15 - 10)
= (15!)/(10! * (15 - 10)!) * (0.8)^10 * 0.2^5
≈ 0.1032

So, the probability that exactly 10 flights arrive on time is approximately 0.1032 or 10.32%.

b. To find the probability that at least 10 flights arrive on time, you need to calculate the probabilities of 10, 11, 12, 13, 14, and 15 flights arriving on time and then sum them up. You can use the same binomial probability formula mentioned above to calculate each individual probability and then add them together.

P(X ≥ 10) = P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15)

Using the provided probabilities table, you can sum up the probabilities for the respective outcomes:

P(X ≥ 10) ≈ 0.1032 + 1.14504E-05 + 9.54204E-07 + 5.50502E-08 + 1.96608E-09 + 3.2768E-11
≈ 0.1032 (rounded to four decimal places)

So, the probability that at least 10 flights arrive on time is approximately 0.1032 or 10.32%.

c. To find the probability that at least 10 flights arrive late, you can use the complement rule. The probability that at least 10 flights arrive late is equal to 1 minus the probability that less than 10 flights arrive late.

P(X ≥ 10 late) = 1 - P(X < 10 late)

P(X < 10 late) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9)

Using the provided probabilities table, you can sum up the probabilities for the respective outcomes:

P(X < 10 late) ≈ 0.035184372 + 0.131941395 + 0.230897442 + 0.250138895 + 0.187604171 + 0.103182294 + 0.042992623 + 0.013819057 + 0.003454764 + 0.00067176
≈ 1 (rounded to four decimal places)

P(X ≥ 10 late) = 1 - P(X < 10 late)
= 1 - 1
= 0

So, the probability that at least 10 flights arrive late is 0.

d. To determine if it would be unusual for Southwest to have 5 flights arrive late, you can compare the probability of this outcome to a threshold value. If the probability is lower than the threshold value, then it would be considered unusual.

Using the provided probabilities table, we find the probability of having exactly 5 flights arrive late:

P(X = 5 late) ≈ 0.103182294

The probability of 5 flights arriving late is greater than 0, indicating that it is a possible outcome. However, without a specific threshold value given, it is difficult to determine if it is considered unusual or not.

Explain Bot Explain Bot answered
8 months ago

To answer the questions:

a. To find the probability that exactly 10 flights arrive on time, we need to calculate the number of ways 10 flights can be on time out of the 15 flights. This can be done using the combination formula, which is n-choose-x, or nCx, where n is the total number of flights and x is the number of flights on time. In this case, n = 15 and x = 10.

The formula for n-choose-x is n!/x!(n-x)!. Plugging in the values, we get 15!/10!(15-10)! = 3003.

Next, we need to calculate the probability of exactly 10 flights arriving on time. This would be the probability of an individual flight arriving on time raised to the power of 10 (since we want 10 flights on time), multiplied by the probability of an individual flight arriving late raised to the power of 5 (since we want 5 flights arriving late).

In this case, the probability of a flight arriving on time is 0.8, and the probability of a flight arriving late is 0.2. So the probability is (0.8)^10 * (0.2)^5 = 0.0000344.

Multiplying this probability by the number of ways we calculated earlier, we get 0.0000344 * 3003 = 0.1032, or 10.32%.

b. To find the probability that at least 10 flights arrive on time, we need to calculate the probabilities for exactly 10, 11, 12, 13, 14, and 15 flights arriving on time, and then sum them up.

Using the methodology from part (a), we can calculate the probabilities of these events happening. I apologize for not completing the calculations, but they can be done using the same formula and probabilities provided.

c. To find the probability that at least 10 flights arrive late, we can subtract the probability of having 0 to 9 flights arriving late from 1 (since the sum of all probabilities should equal 1).

Using the methodology from part (a), we can calculate the probabilities of having 0 to 9 flights arriving late, and then subtract them from 1.

d. To determine if it would be unusual for Southwest to have 5 flights arrive late, we can compare the probability of this event happening to a certain threshold. In statistics, a commonly used threshold for determining unusual events is a significance level of 0.05, which corresponds to the probability of 5% or lower.

By calculating the probability of having exactly 5 flights arrive late using the methodology from part (a) and comparing it to 0.05, we can determine if it would be considered unusual or not.

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