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I was told the answer was 2/5, but I don't understand how they got this.

Here's what I worked out:

A = last toss is tails

B = 30 of the 50 tosses are heads

P(A|B) = P(AB) / P(B)

P(AB) = (1/2)^50 * (49 choose 29)

P(B) = (1/2)^50 * (50 choose 30)

which gives me 3/5.

  • 1
    If $30$ out of $50$ are *heads*, then any individual toss is more likely to have been heads than tails. So the probability that the last is *tails* should be less than $\frac12$. Your "49 choose 29" should be "49 choose 30" or "49 choose 19" – 2017-02-07
  • 1
    As there is nothing special about the last toss, this is the same as asking: what is the probability of drawing a red ball from an urn that contains $20$ red balls and $30$ blue balls (and no others). – 2017-02-07

2 Answers 2

1

Intuitively you can think of an urn with $30$ heads and $20$ tails in it. You draw the coins one by one. The chance the last one is a tail is the same as the chance the first one is a tail and is $\frac 25$.

In your calculation $49 \choose 29$ is the number of ways to have the last one heads, not tails. That is why you get $\frac 35$, not $\frac 25$. You should have done $\frac {{49 \choose 19}}{{50 \choose 20}}=\frac 25$

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It should be P(AB) = (1/2)^50 * (49 choose 30) since you know that the last is tail, so the 30 heads must be within the remaining 49. Then the result 2/5 follows.