A die is rolled until the first time [math]T[/math] that a six turns up. Find [math]P(T \gt 6 | T \gt 3)[/math].

• 0.58
• 0.60
• 0.62
• 0.65
• 0.67

References

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

Solution: A

We have

[[math]]P(T\gt6|T\gt3) = \frac{P(T\gt6)}{P(T\gt3)}[[/math]]

. However, [math]T[/math] has a geometric distribution with [math]P(T=k) = (5/6)^{k-1}(1/6) [/math] which means that

[[math]] \frac{P(T\gt6)}{P(T\gt3)} = \frac{\sum_{k\geq 7} (5/6)^{k-1}}{\sum_{k \geq 4} (5/6)^{k-1}} = \frac{(5/6)^6}{(5/6)^3} = (5/6)^3 = 0.5787. [[/math]]