A die is rolled until the first time $T$ that a six turns up. Find $P(T>6|T>3)$.

• 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

$$P(T>6|T>3)=\frac{P(T>6)}{P(T>3)}$$

. However, $T$ has distribution $P(T=k)=(5/6{)}^{k-1}(1/6)$ which means that

$$\frac{P(T>6)}{P(T>3)}=\frac{\sum _{k\ge 7}(5/6{)}^{k-1}}{\sum _{k\ge 4}(5/6{)}^{k-1}}=\frac{(5/6{)}^6}{(5/6{)}^3}=(5/6{)}^3=0.5787.$$