Three random bonuses all small

Question

A game awards three independent bonuses, each equally likely to be anywhere between 0 and 100 points (a uniform draw). The three draws do not influence one another.

What is the probability that all three bonuses come out below 50 points?

Accepted Answer

Start with a single draw. For one uniform value on 0 to 100, the chance of landing below 50 is just the proportion of the range that lies below 50: P(	ext{one draw} < 50) = \frac{50}{100} = \frac12. Because the three draws are independent, the chance that all three clear the same hurdle is the product of the individual chances (independent "and" means multiply): P(	ext{all three} < 50) = \left(\frac12ight)^3 = \frac18 = 0.125. Notice this is the same as asking for the probability that the largest of the three draws is below 50, since "all below 50" and "the maximum is below 50" describe t…

Three random bonuses all small

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