Q 1) Use Binomial Probabilities Table VII from Appendix section A-22(attached)
to find probability that for the certain number of trials N
and probability of success in each trial p
number of successes will be:
1) exactly k
2) less than k
3) more than k
For N, p, k use numbers assigned= N= 12, K=5 p=0.4
Table below shows number of people who work for the company,
separated by age and gender.
For M1,M2,F1,F2 use numbers assigned for you= M1=35,M2=25, F1=60, F2=40
Based on data in your table find different probability:
1) that randomly selected person is Female and is less than 40 years old: P( Female and Below 40).
2) Probability that randomly selected person is Male or person is above 40 years old: P (Male or Above 40)
3) Find the probability that first selected person is male above 40 and second person is female with age below 40.
(1) Given N=12, p=.4 and k=5:
(a) From the binomial distribution table P(x=5) is 0.227. We can calculate this directly as `_12C_5 (.4)^5(.6)^7 ` using `P(x=k)= ` ` ``_NC_k (p)^k(q)^(N-k) ` where N is the total of subjects, k is the number chosen from the group, p is the probability of an individual having the sought after characteristic, and q is the complement of p.
Here `_(12)C_5 (.4)^5(.6)^7=792(.01024)(.0279936)~~.227 `
(b) To find P(x<5) we sum P(x=0)+P(x=1)+P(x=2)+P(x=3)+P(x=4).
From the table add .002+.017+.064+.142+.213=.438
(c) To find P(x>5) we could add P(x=6)+ ... +P(x=12). Alternatively, we can realize that P(x>5) is the complement of `P(x<=5)=P(x=5)+P(x<5)=.227+.438=.665 `
Check: from the table add .177+.101+.042+.012+.002=.334. The table does not have entries for k=11 or 12 which accounts for the discrepancy. My calculator gives .3347914423
(2) There are 160 total employees: 60 male and 100 female. 95 employees are less than 40, while 65 employees are older than 40.
(a) Find P(F and <40) -- there are 60 females that are younger than 40, so the probability is `60/160=3/8 ` .
Alternatively, the probability of being younger than 40 and female is `(95/160)*(60/95)=60/160=3/8 ` (95/160 is the probability of being younger than 40 while 60/95 is the probability of being female given that you are under 40.
(b) Find P(M or >40). These events are not disjoint (or not mutually exclusive) so we use `P(A "or"B)=P(A)+P(B)-P(Ann B)`
Here P(M)=60/160 and P(>40)=65/160 while the intersection is 25/160 so:
P(M or >40)=`60/160+65/160-25/160=5/8 `
Check: There are 60 males and 65 employees over 40; 25 of those employees were already counted so we have 60+40=100 employees that are male or over 40 and the probability is `100/160=5/8 `
(c) Selecting two people, find P(f,<40|m,>40):
These are not independent events, assuming that the male is removed from the list after selection.
P(M>40)=25/160=5/32. Then P(F<40)=60/159 (there are 60 females under 40 to choose, but only 159 people to choose from after removing the male.)