HMMT 十一月 2010 · GEN1 赛 · 第 1 题

英文解析

1. [ 2 ] Jacob flips five coins, exactly three of which land heads. What is the probability that the first twoare both heads?
   Answer: We can associate with each sequence of coin flips a unique word where H represents 3
   heads, and T represents tails. For example, the word HHTTH would correspond to the coin flip 10
   sequence where the first two flips were heads, the next two were tails, and the last was heads. We aregiven that exactly three of the five coin flips came up heads, so our word must be some rearrangementof HHHTT. To calculate the total number of possibilities, any rearrangement corresponds to a choice
   ( )
   of three spots to place the H flips, so there are = 10 possibilities. If the first two flips are both 5
   heads, then we can only rearrange the last three HTT flips, which corresponds to choosing one spot 3
   ( )
   for the remaining H. This can be done in = 3 ways. Finally, the probability is the quotient of 3
   these two, so we get the answer of . Alternatively, since the total number of possiblities is small, we 31
   can write out all rearrangements: HHHTT, HHTHT, HHTTH, HTHHT, HTHTH, HTTHH, THHHT,10
   THHTH, THTHH, TTHHH. Of these ten, only in the first three do we flip heads the first two times,
   so we get the same answer of .3
   10