One coin in a collection of 65 has two heads. The rest are fair. If a coin, chosen at random from the lot and then tossed, turns up heads 6 times in a row, what is the probability that it is the two-headed coin?

• 0.4
• 0.45
• 0.5
• 0.55
• 0.6

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

The probability that a coin is in the [math]i[/math]th box is [math]1/(i+1)[/math]. If you search in the [math]i[/math]th box and it is there, you find it with probability [math]i/(1+i)[/math]. Determine the probability that the coin is in second box, given that you have looked in the fourth box and not found it.

• 0.35
• 0.4
• 0.45
• 0.5
• 0.55

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

A blood test indicates the presence of a particular disease 95% of the time when the disease is actually present. The same test indicates the presence of the disease 0.5% of the time when the disease is not actually present. One percent of the population actually has the disease.

Calculate the probability that a person actually has the disease given that the test indicates the presence of the disease.

• 0.324
• 0.657
• 0.945
• 0.950
• 0.995

(Chung^{[Notes 1]}) In London, half of the days have some rain. The weather forecaster is correct 2/3 of the time, i.e., the probability that it rains, given that she has predicted rain, and the probability that it does not rain, given that she has predicted that it won't rain, are both equal to 2/3. When rain is forecast, Mr. Pickwick takes his umbrella. When rain is not forecast, he takes it with probability 1/3. Find the probability that he brings his umbrella, given that it doesn't rain.

• 0
• 1/3
• 1/2
• 2/3
• 7/9

Notes

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.