Geometric probabilities solution verificationAn unbiased coin is tossed until a head appears and then tossed until a tail appears. If the tosses are independent, what is the probability that a total of exactly n tosses will be required?My attempt:P(n tosses required to produce one head and one tail) = P ( x tosses needed for first head ) × P ( y tosses needed for first tail ) where x + y = n .So, the probability becomes ( 1 2 ) x − 1 ⋅ ( 1 2 ) ⋅ ( 1 2 ) y − 1 ⋅ ( 1 2 ) = ( 1 2 ) n .This is not the correct answer, however. The correct answer is ( 1 2 ) n ⋅ ( n − 1 ). Can someone pleas explain what I did incorrectly and where the n − 1 factor is coming from?

Question

Geometric probabilities solution verificationAn unbiased coin is tossed until a head appears and then tossed until a tail appears. If the tosses are independent, what is the probability that a total of exactly n tosses will be required?My attempt:P(n tosses required to produce one head and one tail)   =  P  (  x tosses needed for first head  )  ×  P  (  y tosses needed for first tail  )  where     x  +  y  =  n  .So, the probability becomes                                           (                          1              2                        )                                x            −            1                          ⋅                  (                      1            2                    )                ⋅                              (                          1              2                        )                                y            −            1                          ⋅                  (                      1            2                    )                =                              (                          1              2                        )                                n                              .This is not the correct answer, however. The correct answer is             (              1        2            )              n        ⋅  (  n  −  1  ). Can someone pleas explain what I did incorrectly and where the   n  −  1 factor is coming from?

Riya Cline · Accepted Answer

Step 1Your thinking is almost there - except in your argument, x could be anything from 1 to   n  −  1 (since getting a head on xth toss and then getting a tail on the   y  =  n  −  x th toss gives you a head and tail on exactly the nth toss). So if we sum your expression over all possible combinations of x and   y  =  n  −  x, we have   n  −  1 times the expression you got.Step 2More generally, this is an example of (a variant of) the Negative Binomial distribution, which counts the number of successes needed for r events to happen.

An unbiased coin is tossed until a head appears and then tossed until a tail appears. If the tosses are independent, what is the probability that a total of exactly n tosses will be required?

Answered question

Answer & Explanation

New Questions in High school geometry