To find the probability that from 9 to 12 mountain climbers reach the summit, including both 9 and 12, you can use the binomial probability formula. In this case, you're looking for the probability of success (reaching the summit) which is 65% or 0.65, and you want to consider the range of 9 to 12 successes in a sample of 16 climbers.
You can calculate this probability for each value in the range and then add them together:
P(9 climbers reach the summit) = C(16, 9) * (0.65^9) * (0.35^7)
P(10 climbers reach the summit) = C(16, 10) * (0.65^10) * (0.35^6)
P(11 climbers reach the summit) = C(16, 11) * (0.65^11) * (0.35^5)
P(12 climbers reach the summit) = C(16, 12) * (0.65^12) * (0.35^4)
Where C(n, k) is the binomial coefficient, which is the number of ways to choose k climbers out of n climbers. You can use a calculator to compute these values.
Now, add these probabilities together:
P(9 to 12 climbers reach the summit) = P(9) + P(10) + P(11) + P(12)
This will give you the probability of having between 9 and 12 climbers reach the summit in the sample of 16. Round the answer to four decimal places.
Let's compute them.
P(9 climbers reach the summit):
[P(9) = \binom{16}{9} times (0.65^9) times (0.35^7)
P(10 climbers reach the summit):
\[P(10) = \binom{16}{10} \times (0.65^10) \times (0.35^6)\]
P(11 climbers reach the summit):
\[P(11) = \binom{16}{11} \times (0.65^11) \times (0.35^5)\]
P(12 climbers reach the summit):
\[P(12) = \binom{16}{12} \times (0.65^12) \times (0.35^4)\]
Now, calculate these probabilities individually and then sum them:
\[P(9 \text{ to } 12 \text{ climbers}) = P(9) + P(10) + P(11) + P(12)\]
After performing the calculations, round the result to four decimal places to find the probability of having between 9 and 12 climbers reach the summit in the sample of 16 climbers.