Question

1. A particular event has a probability of occurrence of p=0.65. Given 12 independent trials, draw the probability distribution and calculate the expected value.
2. A particular event has a probability of occurrence of p=0.65. If each outcome is independent of the last, draw the probability distribution associated with the event not occurring a given number of consecutive times and calculate the expected value.
3. Given the following scenario, determine which distribution (binomial, geometric, hypergeometric) is the most appropriate to model it? Rank the distributions from most appropriate to least appropriate. Explain the reason(s). "1600 students attend BOSS. It is determined that 70% of all BOSS students wear glasses. If groups of 50 students were selected at random, how many of them would be wearing glasses?"

Answer 1

a) The number of occurrences out of the 12 independent trials could be modelled here as:

The probabilities here are computed as:

X	p(x)
0	3.37922E-06
1	7.53083E-05
2	0.000769221
3	0.004761844
4	0.019897705
5	0.059124608
6	0.128103318
7	0.203919567
8	0.236692354
9	0.195365118
10	0.10884628
11	0.036753289
12	0.005688009

This is plotted in a graph as:

b) Now here we are finding the distribution of the variable representing the event not happening for some consecutive number of times:

P(X = x) = (1 - 0.65)x*0.65

x	p(x)
0	0.65
1	0.2275
2	0.079625
3	0.02786875
4	0.00975406
5	0.00341392
6	0.00119487
7	0.00041821
8	0.00014637
9	5.123E-05
10	1.7931E-05
11	6.2757E-06
12	2.1965E-06
13	7.6877E-07
14	2.6907E-07
15	9.4175E-08

This is plotted in graph as:

c) 1600 students attend BOSS. It is determined that 70% of all BOSS students wear glasses. If groups of 50 students were selected at random, the number of them expected to be wearing glasses is computed here as:

This is computed using the binomial distribution here.