Question

72% of all students at a college still need to take another math class. If 36 students are randomly selected, find the following probabalities. Round your answers to 4 decimal places.

a. Exactly 26 of them need to take another math class.
b. At most 25 of them need to take another math class.
c. At least 24 of them need to take another math class.
d. Between 25 and 32 (including 25 and 32) of them need to take another math class.

Answer 1

Solution-:

Let, X= Number of students at a college still need to take another math class among 36 students.

Given:

This situation is suitable for binomial distribution with parameter n=36 and p=0.72

The p.m.f. of X is given by,

  

We prepare the following table for finding probabilities:

	n=36	p=0.72	q=0.28
x	nCx	p^x	q^(n-x)	P[X=x]
0	1	1.0000	0.0000	0.0000
1	36	0.7200	0.0000	0.0000
2	630	0.5184	0.0000	0.0000
3	7140	0.3732	0.0000	0.0000
4	58905	0.2687	0.0000	0.0000
5	376992	0.1935	0.0000	0.0000
6	1947792	0.1393	0.0000	0.0000
7	8347680	0.1003	0.0000	0.0000
8	30260340	0.0722	0.0000	0.0000
9	94143280	0.0520	0.0000	0.0000
10	254186856	0.0374	0.0000	0.0000
11	600805296	0.0270	0.0000	0.0000
12	1251677700	0.0194	0.0000	0.0000
13	2310789600	0.0140	0.0000	0.0000
14	3796297200	0.0101	0.0000	0.0000
15	5567902560	0.0072	0.0000	0.0001
16	7307872110	0.0052	0.0000	0.0003
17	8597496600	0.0038	0.0000	0.0010
18	9075135300	0.0027	0.0000	0.0027
19	8597496600	0.0019	0.0000	0.0067
20	7307872110	0.0014	0.0000	0.0146
21	5567902560	0.0010	0.0000	0.0286
22	3796297200	0.0007	0.0000	0.0502
23	2310789600	0.0005	0.0000	0.0786
24	1251677700	0.0004	0.0000	0.1095
25	600805296	0.0003	0.0000	0.1351
26	254186856	0.0002	0.0000	0.1470
27	94143280	0.0001	0.0000	0.1400
28	30260340	0.0001	0.0000	0.1157
29	8347680	0.0001	0.0001	0.0821
30	1947792	0.0001	0.0005	0.0493
31	376992	0.0000	0.0017	0.0245
32	58905	0.0000	0.0061	0.0098
33	7140	0.0000	0.0220	0.0031
34	630	0.0000	0.0784	0.0007
35	36	0.0000	0.2800	0.0001
36	1	0.0000	1.0000	0.0000

(a) P[Exactly 26 of them need to take another math class]

The required probability is 0.1470

(b) P[At most 25 of them need to take another math class]

(From above table)

The required probability is 0.4276.

(c) P[At least 24 of them need to take another math class]

Where,

  (From above table)

Then,

P[At least 24 of them need to take another math class]

  

The required probability is 0.8164.

(d) P[Between 25 and 32 (including 25 and 32) of them need to take another math class]

  

(From above table)

The required probability is 0.7036.