Problem Solving

VJesus12 wrote:The high temperature of a summer day on Fire Island is always one of three temperatures: 84, 85 or 86 degrees. If there is an equal chance of each temperature, what is the probability that the median high temperature over a three day period will be 85 degrees?

A. 1/27
B. 2/9
C. 1/3
D. 4/9
E. 13/27

[spoiler]OA=E[/spoiler]

Source: Princeton Review

Since each day has 3 possible outcomes for the high temperature, 3 consecutive days have 3 x 3 x 3 = 27 total possible outcomes for the high temperature. The number of ways to get 85 as a median can occur in the following ways:

One reading of 84, two readings of 85: By the permutations of indistinguishable objects formula, this can occur in 3!/2! = 3 ways. (which are (84, 84, 85), (84, 85, 84); (85, 84, 84)).

One reading of 86, two readings of 85: This can also occur in 3 ways, as explained above.

One reading of each temperature: This can occur in 3! = 6 ways.

Three readings of 85: This can occur in one way.

Thus, a median high temperature of 85 degrees occurs in 3 + 3 + 6 + 1 = 13 outcomes. Thus, the probability is 13/27.

Answer: E