You have two coins, a blue and a red one. You choose one of the coins at random,

Question

You have two coins, a blue and a red one. You choose one of the coins at random, each being chosen with probability 1/2. You then toss the chosen coin twice. The coins are biased: with the blue coin, the probability of heads in any given toss is 0.8, whereas for the red coin it is 0.2.

• Let B denote the event that you’ve picked the blue coin.
• Let H i denote the event that the i-th toss resulted in heads for i = {1,2}.

Notice that given the choice of a coin, events H 1 and H 2 are independent,
by assumption.

(a) Compute P(H1 ).
(b) Compute P(H2 ).
(c) Compute P(H1 B).
(d) Compute P(H1 H2 ).
(e) Are events H1 and H2 independent? Why?
(f) Compute P(H2 | H1 ).

Explanation / Answer

a)P(H1) =P(B)*P(H1|B)+P(R)*P(H1|R) =0.5*0.8+0.5*0.2=0.5

b)P(H2) =P(H1) =0.5 (as above)

c) Compute P(H1 B) =P(B)*P(H1|B) =0.5*0.8 =0.4

d) Compute P(H1 H2 ) =P(B)*P(H1nH2)/P(B)+P(R)*P(H1nH2)/P(R)=0.5*0.8*0.8+0.5*0.2*0.2=0.34

e) as P(H1nH2) is not equal to P(H1)*P(H2) ; therefore they are not independent.

f)P(H2|H1) =P(H1nH2)/P(H1) =0.34/0.5=0.68

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