# Probaility

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

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