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If the probability that the Islanders will beat the Rangers in a game is 1//6, what is the probability that the Islanders will win exactly five out of six games in a series against the Rangers? Round your answer to the nearest thousandth. Answer:

If the probability that the Islanders will beat the Rangers in a game is 1/6 1 / 6 , what is the probability that the Islanders will win exactly five out of six games in a series against the Rangers? Round your answer to the nearest thousandth.\newlineAnswer:

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Q. If the probability that the Islanders will beat the Rangers in a game is 1/6 1 / 6 , what is the probability that the Islanders will win exactly five out of six games in a series against the Rangers? Round your answer to the nearest thousandth.\newlineAnswer:
  1. Understand Problem and Formula: Understand the problem and determine the formula to use.\newlineWe need to calculate the probability of the Islanders winning exactly five out of six games. This is a binomial probability problem, where the number of trials is 66, the number of successes is 55, and the probability of success on a single trial is 16\frac{1}{6}. The formula for the probability of exactly kk successes in nn trials is:\newlineP(X=k)=(nk)⋅(pk)⋅((1−p)(n−k))P(X = k) = \binom{n}{k} \cdot (p^k) \cdot ((1-p)^{(n-k)})\newlinewhere (nk)\binom{n}{k} is the binomial coefficient, pp is the probability of success, and (1−p)(1-p) is the probability of failure.
  2. Calculate Binomial Coefficient: Calculate the binomial coefficient for 55 successes in 66 trials.\newlineThe binomial coefficient (nn choose kk) is calculated as:\newline(n(n choose kk) = n!k!⋅(n−k)!\frac{n!}{k! \cdot (n-k)!}\newlineFor our problem, this is:\newline(6(6 choose 55) = 6!5!⋅(6−5)!\frac{6!}{5! \cdot (6-5)!}\newline= 6600\newline= 66
  3. Calculate Probability of 55 Successes: Calculate the probability of exactly 55 successes (wins) and 11 failure (loss).\newlineUsing the formula from Step 11, we have:\newlineP(X=5)=(65)⋅(16)5⋅(56)1P(X = 5) = \binom{6}{5} \cdot \left(\frac{1}{6}\right)^5 \cdot \left(\frac{5}{6}\right)^1\newlineNow we plug in the values from Step 22:\newlineP(X=5)=6⋅(16)5⋅(56)1P(X = 5) = 6 \cdot \left(\frac{1}{6}\right)^5 \cdot \left(\frac{5}{6}\right)^1
  4. Perform Calculations: Perform the calculations.\newlineP(X=5)=6×(16)5×(56)P(X = 5) = 6 \times \left(\frac{1}{6}\right)^5 \times \left(\frac{5}{6}\right)\newline=6×(17776)×(56)= 6 \times \left(\frac{1}{7776}\right) \times \left(\frac{5}{6}\right)\newline=6×57776= 6 \times \frac{5}{7776}\newline=307776= \frac{30}{7776}\newline=1259= \frac{1}{259}
  5. Convert Probability to Decimal: Convert the probability to a decimal and round to the nearest thousandth.\newline1259\frac{1}{259} is approximately equal to 0.003860.00386 when converted to a decimal.\newlineRounded to the nearest thousandth, this is 0.0040.004.

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