A box contains 10 tickets, marked 1 through 10. 1 ticket is drawn at random from
ID: 3217048 • Letter: A
Question
A box contains 10 tickets, marked 1 through 10. 1 ticket is drawn at random from the box.
Two possible outcomes will be looked at: A and B, listed below.
(a) Find P(A) and P(B); are A and B independent? Mutually exclusive?
(b) Find P(both A and B). Would multiplying the unconditional probabilities, P(A) and P(B), give the correct answer, or are conditional probabilities required? Why or why not?
(c) Find P(at least one of A or B)=P(A or B). Would adding the individual probabilities of A and B give the correct answer? Why or why not?
(d) Find P(B|A) and decide if it is equal to P(B). Based on this decision, are A and B independent?
A: The ticket drawn has an even number on it.
B: The number on the ticket drawn is less than 6.
Explanation / Answer
Box contains 10 tickets names 1 to 10
A: The ticket drawn has an even number on it.
B: The number on the ticket drawn is less than 6
P(A) = 5/10 = 1/2 = 0.5 (for wven numbers 2,4,6,8,10)
P(B) = 5/10 = 0.5 (for numbers less than 6 i.e 5,4,3,2,1)
Independent Events:
P(A and B) = P(A) × P(B)
P(A and B) = 2/10 = 1/5
P(A) × P(B) = 1/2 * 1/2 = 1/4
As P(A and B) does not equal P(A) × P(B) hence they are independent
b)
P(both A and B). = 2/10 (as a number even and less than 6 are 2 and 4)
multiplying the unconditional probabilities, P(A) and P(B) would not give the correct answer as the are not independent.
c)
Find P(at least one of A or B)= 8/10 = 0.8 (as for atleast one of A or B there are 8 outcomes 1,2,3,4,5,6,8,10 )
No adding the individual probabilities of A and B give the correct answer as there are some outcome which are same in both the outcomes (outcome of 2 and 4)
d)
P(B|A) = P(A) * P(B) / P(A) = (0.5*0.5 )/ 0.5 = 0.5
Yes it is equal to P(B)
No A and B are not independent
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