0
$\begingroup$

Suppose $20$ tosses of a coin yield $8$ heads, $12$ tails.

Let $X$ be the # of sequences of a head followed by exactly $2$ tails.

Let $Y$ be the # of sequences of a head followed by at least $2$ tails.

What are the values of $E[X]$ and $E[Y]$ ?

My efforts are meeting dead ends.

  • 0
    @utdiscant: One or more HTT in the sequence. Needn't start with HTT.2012-08-12

1 Answers 1

1
  1. The probability that HTTH will appear in a particular position is $\dfrac{16 \choose 6}{20 \choose 8} = \dfrac{308}{4845}$ and there are 17 possible positions so $E[X] = \frac{308}{285} \approx 1.0807.$

  2. The probability that HTT will appear in a particular position is $\dfrac{17 \choose 7}{20 \choose 8} = \dfrac{44}{285}$ and there are 18 possible positions so $E[Y] = \frac{264}{95} \approx 2.7789.$

  • 0
    Thanks, I was lost trying to compute the # of cases with 1,2... favorable sequences when only the expectation was asked for !2012-08-12