Question

Identify the distribution and give a symbolic expression for each indicated probability, identifying parameters. Then use Mathematica or Excel to evaluate the indicated probabilities.

1. In a large biochemistry class of 41 chemistry and 81 biology majors, 7 students are selected at random to prepare a presentation. What is the probability that
   1. 2 chem and 5 bio students are selected? (b) At least one chemistry major is selected?
2. Blood tests of winning horses in thoroughbred races disqualify 9% of them for use of illegal medications. What is the likelihood that
   1. No horse is disqualified in this weekend’s 19 races?
   2. The first disqualified winning horse occurs between the 23^rd and 31^st race (inclusive)?
3. During rush hour, cars pass a given point on the parkway at the instantaneous rate of 98 per minute.
   1. What is the probability that at least 430 pass that point in a five-minute period?
4. In a state that requires licenses for Physician Assistants, 39% of applicants pass the licensing exam on the first try. If 2680 students took the exam for the first time the last time it was administered,
   1. What is the likelihood that at most 1000 passed?
5. 6.25% of logins to the Seton Hall website fail. Assuming attempts are independent:
   1. What is the probability that the 10^th failure occurs on the 140^th login?
   2. What is the probability that the 6^th failure has not occurred in the first 100 logins?
6. Individuals who barely survive major disasters, on average, suffer nightmares during sleep every 2.75 hours [during the first month].
   1. Find the probability that someone’s 10^th nightmare occurs within the first 40 hours of sleep.

1. For each of the derivations above, give the mean and standard deviation. Use the formulas from the notes. For Geometric and Negative Binomial, be careful about variations.

Answer 1

1. Chemistry -41

Biology -81

Total = 41+81= 122

(a) Total number of cases of selecting 7 students =

Favorable number of cases = *

Probability of selecting 2 chem and 5 bio students= favorable number of cases / total number of cases

= *

= 0.3137

Note : using excel " =COMBIN(41,2)*COMBIN(81,5)/COMBIN(122,7)"

(b) Out of 7 selected ,

If chem- 1 , then bio -6

If chem 2 , then bio - 5

.....

If chem -7 , bio -0

Probability of selecting atleast one chem major

=   * + ....... +   *

= 0.9481

chem	bio	Probability
1	6	0.1986477
2	5	0.3136543
3	4	0.2647731
4	3	0.128992
5	2	0.0362484
6	1	0.0054373
7	0	0.0003356
		0.9480886

formula for each probability is calculated as " =COMBIN(41,A13)*COMBIN(81,B13)/COMBIN(122,7)"

where A13 cell represents chem, B13 cell represents bio

2. (a)Using Binomial probability law

Let X be the number of horses disqualified

X follow Binomial with n = 19 , p = 0.09

P(X=0) =0.1666

Note :Using excel " =BINOM.DIST(0,19,0.09,FALSE)"

(b) To find the probability that first disqualified horse in is 23 rd race

We first find no disqualified horse in 22 races

and then joint probability of no disqualified horse in 22 races and probability of disqualifying a horse

n		P(X=0)	P(X=0)*0.09
22		0.125577	0.01130194
23		0.114275	0.01028477
24		0.10399	0.00935914
25		0.094631	0.00851682
26		0.086114	0.0077503
27		0.078364	0.00705278
28		0.071311	0.00641803
29		0.064893	0.0058404
30		0.059053	0.00531477
			0.07183895

Thus required probability = 0.0718