# Solved – I have 2 problems with conditional probability, involving coins and dice

Hi I have these problems:

``A coin is tossed three times. What is the probability that exactly two heads occur, given that (a) the ﬁrst outcome was a head? (b) the ﬁrst outcome was a tail? (c) the ﬁrst two outcomes were heads? (d) the ﬁrst two outcomes were tails? (e) the ﬁrst outcome was a head and the third outcome was a head? ``

Which I have done it like:

``T1 T2 T3  H  H  H H  H  T H  T  H H  T  T T  H  H T  H  T T  T  H  T  T  T ``

I put all the outcomes, then by counting them by hand:

``p(2H|T1=H)=2/4 p(2H|T1=T)=1/4 p(2H|T1=H,T2=H)=1/2 p(2H|T1=T,T2=T)=0 p(2H|T1=H,T3=H)=1/2 ``

Is this correct? Then, is there another way of doing it? Counting by hand just feel wrong, there must be a formal way.

Then:

``A die is rolled twice. What is the probability that the sum of the faces is greater than 7, given that (a) the ﬁrst outcome was a 4? (b) the ﬁrst outcome was greater than 3? (c) the ﬁrst outcome was a 1? (d) the ﬁrst outcome was less than 5? ``

Again counting, this really feels so wrong, I'm expecting some formula to get them.

``p(sum>7|D1=4)=3/6 p(sum>7|D1>3)=12/18 p(sum>7|D1=1)=0/6 p(sum>7|D1<5)=6/24 ``

Am I correct? Thanks.

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