Question

According to an​ airline, a particular flight is on time 88​% of the time. Suppose 33 flights are randomly selected and the number of on time flights is recorded. Find the probabilities of the following events occurring.

a. All 33 flights are on time

b. Between 25 and 27 flights​ (inclusive) are on time

Answer 1

Let X be the number of flights are on time

Here , X has binomial distribution with parameter n=33 and p=0.88

Now , the PMF of X is ,

; x=0,1,2,3..............,n and q=1-p

= 0 ; otherwise

The table of probability distribution is ,

X				P(X=x)
0	1	1	4.1E-31	4E-31
1	33	0.88	3.4E-30	1E-28
2	528	0.7744	2.8E-29	1E-26
3	5456	0.68147	2.4E-28	9E-25
4	40920	0.5997	2E-27	5E-23
5	237336	0.52773	1.6E-26	2E-21
6	1107568	0.4644	1.4E-25	7E-20
7	4272048	0.40868	1.1E-24	2E-18
8	1.4E+07	0.35963	9.5E-24	5E-17
9	3.9E+07	0.31648	7.9E-23	1E-15
10	9.3E+07	0.2785	6.6E-22	2E-14
11	1.9E+08	0.24508	5.5E-21	3E-13
12	3.5E+08	0.21567	4.6E-20	4E-12
13	5.7E+08	0.18979	3.8E-19	4E-11
14	8.2E+08	0.16702	3.2E-18	4E-10
15	1E+09	0.14697	2.7E-17	4E-09
16	1.2E+09	0.12934	2.2E-16	3E-08
17	1.2E+09	0.11382	1.8E-15	2E-07
18	1E+09	0.10016	1.5E-14	2E-06
19	8.2E+08	0.08814	1.3E-13	9E-06
20	5.7E+08	0.07756	1.1E-12	5E-05
21	3.5E+08	0.06826	8.9E-12	0.0002
22	1.9E+08	0.06006	7.4E-11	0.0009
23	9.3E+07	0.05286	6.2E-10	0.003
24	3.9E+07	0.04651	5.2E-09	0.0093
25	1.4E+07	0.04093	4.3E-08	0.0244
26	4272048	0.03602	3.6E-07	0.0551
27	1107568	0.0317	3E-06	0.1048
28	237336	0.02789	2.5E-05	0.1647
29	40920	0.02455	0.00021	0.2083
30	5456	0.0216	0.00173	0.2037
31	528	0.01901	0.0144	0.1445
32	33	0.01673	0.12	0.0662
33	1	0.01472	1	0.0147

a. Now ,

P(X=33)=0.0147

Therefore , the probability of all 33 flights are on time is 0.0147

b. Now ,

Therefore , the probability between 25 and 27 flights are on time is 0.1843