Suppose you are eating at cafeteria with two friends. You have agreed to the following rule to decide who will pay the bill. Each person will toss a coin. The person who gets a result that is different from the other two will pay the bill. If all the three tosses yield the same result the bill will be shared by all. Find the probability that

1. Suppose you are eating at cafeteria with two friends. You have
agreed to the following rule to decide who will pay the bill.
Each person will toss a coin. The person who gets a result that is
different from the other two will pay the bill. If all the three
tosses yield the same result the bill will be shared by all. Find
the probability that
(a) Only you have to pay
(b) All three will share.
Solution
𝑺 = {𝑯𝑯𝑯, 𝑯𝑯𝑻, 𝑯𝑻𝑯, 𝑯𝑻𝑻, 𝑻𝑯𝑯, 𝑻𝑯𝑻, 𝑻𝑻𝑯, 𝑻𝑻𝑻}
𝒏(𝑺) = 𝟖
(a) Probability of each person to pay the bill is
𝟏
𝟑
.
Let A be the event that any one’s result is different
from others.
𝑨 = {𝑯𝑯𝑻, 𝑯𝑻𝑯, 𝑯𝑻𝑻, 𝑻𝑯𝑯, 𝑻𝑯𝑻, 𝑻𝑻𝑯}
𝒏(𝑨) = 𝟔

2. 𝑷(𝑨) =
𝒏(𝑨)
𝒏(𝑺)
=
𝟔
𝟖
𝑵𝒐𝒘 𝒕𝒉𝒆 𝒑𝒓𝒐𝒃𝒂𝒃𝒊𝒍𝒊𝒕𝒚 𝒕𝒉𝒂𝒕 𝒐𝒏𝒍𝒚 𝒚𝒐𝒖 𝒉𝒂𝒗𝒆 𝒕𝒐 𝒑𝒂𝒚 𝒕𝒉𝒆 𝒃𝒊𝒍𝒍
= 𝑷(𝑨) ×
𝟏
𝟑
=
𝟔
𝟖
×
𝟏
𝟑
= 𝟎. 𝟐𝟓 = 𝟐𝟓%
There is 25% chance, that only you will pay the bill.
(b) Let B be the event that all three will share the bill.
If all the three tosses yield the same result the bill will
be shared by all.
𝑩 = {𝑯𝑯𝑯, 𝑻𝑻𝑻}
𝒏(𝑩) = 𝟐
𝑷(𝑩) =
𝒏(𝑩)
𝒏(𝑺)
=
𝟐
𝟖
= 𝟎. 𝟐𝟓 = 𝟐𝟓%
There is 25% chance, that bill will be shared by all.