Question

Two fair coins and a fair die are tossed. Find the sample space of the
experiment (10 pts); Find the probabilities of the following events:
A- ”the die shows 2 or 3” (5 pts);
B- ”one of the coins is head, the other - tail, and the die shows an odd number” (5
pts).
Are the events A and B independent? (5 pts).
Give proofs.

Answer 1

solution:

When two fair coins tossed,we have

sample space s1 = { HH,HT,TH,TT }

when a die is thrown, we have

sample space s2 = {1,2,3,4,5,6}

In the event Two fair coins and a fair die is Tossed.we have the following samle space

Sample space S = { HH1,HH2,HH3,HH4,HH5,HH6,

HT1,HT2,HT3,HT4,HT5,HT6,

TH1,TH2,TH3,TH4,TH5,TH6,

TT1,TT2,TT3,TT4,TT5,TT6 }

n(S) = 24

a) Let A = event that die shows 2 or 3 = {HH2,HH3,HT2,HT3,TH2,TH3,TT2,TT3 }

n(A) = 8

P(  die shows 2 or 3 ) = P(A)

= n(A) / n(S)

= 8 / 24

= 0.33

Probability that  die shows 2 or 3 = 0.3333

b) Let B = one of the coins is head, the other - tail, and the die shows an odd number

= { HT1,HT3,HT5,TH1,TH3,TH5 }

n(B) = 6

     P (one of the coins is head, the other - tail, and the die shows an odd number )= P(B)

= n(B) / n(S)

= 6 / 24

= 0.25

Probability that  one of the coins is head, the other - tail, and the die shows an odd number = 0.25

c) If two events A and B are independent then P(A|B) = P(A)

i.e., P(AB) = P(A) * P(B)

Here, AB = {HH2,HH3,HT2,HT3,TH2,TH3,TT2,TT3 } { HT1,HT3,HT5,TH1,TH3,TH5 }

= { HT3,TH3 }

n( AB) = 2

  P(AB) = n( AB) / n(S) = 2/24 = 1/12 =0.0833

P(A) * P(B) = 0.3333 * 0.25

= 0.0833

=  P(AB)

     P(AB) = P(A) * P(B)

A and B are independent events