Rap:
Start with two coins, a fair coin and a biased coin. The probability of picking either coin is 50:50.
Start with fair coin,
1 head =prob of fair coin (1/2)* prob of head(1/2)=1/4
2 heads=fair coin(1/2)*prob of two heads(1/2*1/2)=1/8
3 heads=fair coin(1/2)*prob of three heads(1/2*1/2*1/2)=1/16
biased coin
1 head=prob of biased coin (1/2)*prob of head(1/1)=1/2
2 heads= the same (1/2)
3 heads=the same (1/2)
Prob of 1 head=3/4
Prob of 2 heads = 5/8
Prob of 3 heads =9/16
1 head prob of fair coin =(1/4)/(3/4)=1/3
2 head prob of fair coin = (1/8)/(5/8)=1/5
3 head prob of fair coin = (1/16)/(9/16)=1/9
Oh! I get it, he tosses a tails the third time---it's the fair coin fer shure!
I can speak for all, when I say that was a particularly well explained answer!
The answer was: 1/3, 1/5, 1
a) There are two events: The selection of a coin -- resulting in the Unfair Coin, "U", or the Fair Coin, "F" -- and the flip -- resulting in heads, "H", or tails, "T", for the Fair Coin, and in one of the two heads, "H1" or "H2", for the Unfair Coin. Each of these events is considered a 50/50 proposition.
Event1 Event2
F H
F T
U H1
U H2
So the conditional probability P(F|H) = P(F and H)/P(H) = (1/4)/(3/4) = 1/3.
b) Similarly,
Event1 Event2 Event3
F H H
F H T
F T H
F T T
U H1 H1
U H1 H2
U H2 H1
U H2 H2
Now P(F|H,H) = P(F and H,H)/P(H,H) = (1/8)/(5/8) = 1/5.
c) It must be the fair coin. P(F|H,H,T) = 1.
BPOLTANK
ETRAE