Question

6) There are 5 boxes. In each box there are 5 tickets labelled with the numbers 1,2,3,4,5. An experiment is to draw one number at random from each of the five boxes. Calculate the following probabilities using a calculator, but show some work.
a) P(no two numbers alike).
b) P(exactly two alike). (example 11234)
c) P(exactly three alike). (example 11123)
d) P(exactly four alike). (example 11112)
e) P(two pair). (example 11223)
f) P(the number 4 comes up at least once)

Answer 1

Total number of ways to draw one number at random from each of the five boxes =

(a) Total number of ways to draw such that no two numbers are alike

= 5*4*3*2*1 = 120

Thus, required probability = = 0.0384

(b) Total number of ways to draw such that exactly two numbers are alike = = 1200

(Two boxes can be selected out of 5 boxes in which the ticket of same numbers are present then any of the 5 numbers for those 2 boxes can be selected, the remaining 3 boxes in 4, 3 and 2 ways such that ticket numbers only from two boxes are alike)

Thus, required probability = = 0.384

(c) Similarly, the number of ways = = 600

Thus, the required probability = = 0.192

(d) The number of ways = = 100

Thus, the required probability = 0.032

(e) The number of ways = 1800

Thus, the required probability = 0.576

(f) P(the number 4 comes up at least once)

= 1 - P(the number 4 does not come in any of the draws)

= 1 - = 0.67232