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Question

Two dice are rolled once. Find the probability of getting such numbers on two dice, whose product is a perfect square.

OR

A game consists of tossing a coin 3 times and noting its outcomes each time. Hanif wins if he gets three heads or three tails, and loses otherwise. Calculate the probability that Hanif will lose the game.

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Solution

The sample space of the given experiment is:
S=⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪
n (S)= 36
Let E be the event that the product of numbers obtained on the upper face is a perfect square
E = {(1,1), (1,4), (2,2), (3,3), (4,1), (4,4), (5,5), (6,6)}
n (E) = 8
P(E)=n(E)n(S)=836=29
OR
The set of possible outcomes of the given experiment are:
S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Let E be the event of getting three heads or three tails.
E = {HHH, TTT}
Probability of winning = P (E)
=n(E)n(S)=28=14
Probability of losing = P(E')
= 1 - P (E)
=114=34
Therefore, the probability that Hanif will lose the game is 34


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