Question

Exercise 1.5. In one type of state lottery 5 distinct numbers are picked fro 1,2,3, 40 uniformly at random. Describe a sample space 2 and a probability measure P to model this experiment. (b) What is the probability that out of the five picked numbers exactly three will be even? Exercise 1.6. We have an urn with 3 green and 4 yellow balls. We ch

For 1.5, I need to do this with and without replacement. Thank you

Answer 1

With replacement:

(a)

All 40 numbers are equally likely and numbers can be repeated so each number out of 5 can be selected in 40 ways. So number of outcomes in sample space will be

Since each outcome is equally likely so probability of each outcome is 1 / 102400000.

(b)

Out of 40 numbers 20 are even and 20 are odd so number of ways of selecting 3 even numbers and 2 odd numbers is

(20*20*20)*(20*20) = 3200000

Therefore the probability of selecting 3 even numbers is

P(even numbers) = 3200000 / 102400000 = 0.03125

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Without replacement:

(a)

All 40 numbers are equally likely and numbers cannot be repeated so number of outcomes in sample space will be

Since each outcome is equally likely so probability of each outcome is 1 / 658008.

(b)

Out of 40 numbers 20 are even and 20 are odd so number of ways of selecting 3 even numbers and 2 odd numbers is

Therefore the probability of selecting 3 even numbers is

P(even numbers) = 216600 / 658008= 0.3292