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Chapter 19: Problem 25

For a gas of six molecules confined to a box, find the probability that (a) all the molecules will be found on one side of the box and (b) half the molecules will be found on each side.

Short Answer

Expert verified
The probability that all the molecules will be found on one side of the box is 0.09375, while the probability that half the molecules will be found on each side of the box is 0.3125.

Step by step solution

01

Apply the formula for Part A

Apply the formula for binomial distribution to Part A. In this case, all the molecules (6) are expected to be found on one side of the box. This is equivalent to a single 'success' in six trials. Therefore, \(k = 1\), \(n = 6\) and the probability of success \(p = 0.5\), as it can be either one side or the other. Replace these into the formula to get the result: \( P(k; n, p) = C(6, 1) * (0.5^1) * ((1-0.5)^(6-1)) \).
02

Compute the result for Part A

Compute the result for Part A. The combination \(C(6, 1)\) is equal to 6. Replace these into the formula to find the probability: \(P = 6 * 0.5 * 0.03125 = 0.09375\).
03

Apply the formula for Part B

Apply the formula for binomial distribution for Part B. In this case, half of the molecules (3 out of 6) are expected to be found in each side of the box, equivalent to three 'successes' in six trials. So, again \(n = 6\), now \(k=3\) and \(p=0.5\). Replace these values into the formula to get the result: \( P(k; n, p) = C(6, 3) * (0.5^3) * ((1-0.5)^(6-3)) \).
04

Compute the result for Part B

Compute the result for Part B. The combination \(C(6, 3)\) is equal to 20. Substitute these into the formula to get the final result: \(P = 20 * 0.125 * 0.125 = 0.3125\).

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