Problem 97
A lottery has 1000 tickets: one ticket wins 500 thousand, 10…
A lottery has $1000$ tickets: one ticket wins $500$ thousand, $10$ tickets win $100$ thousand each, $50$ tickets win $20$ thousand each, $100$ tickets win $5$ thousand dinars each, and the remaining tickets win nothing. Someone bought one ticket. What is the probability of winning at least $20$ thousand dinars?
Explanation of Method
"At least $20$ thousand" means a prize of $500$, $100$ or $20$ thousand. These three events are mutually exclusive (a ticket has only one prize), so by the addition rule their probabilities add up. Each probability is the number of such tickets divided by $1000$.
Step 1: The events
| Event | Prize | Tickets | Probability |
|---|---|---|---|
| $A$ | $500$ thousand | $1$ | $0.001$ |
| $B$ | $100$ thousand | $10$ | $0.010$ |
| $C$ | $20$ thousand | $50$ | $0.050$ |
| $D$ | $5$ thousand | $100$ | $0.100$ |
| — | nothing | $839$ | $0.839$ |
Step 2: The addition rule
The events $A$, $B$, $C$ are mutually exclusive, so
\[ P(A+B+C) = P(A)+P(B)+P(C) = 0.001+0.010+0.050. \]
\[ \boxed{P = 0.061.} \]
Step 3: Remarks
- The probability of winning anything is $0.001+0.01+0.05+0.1=0.161$.
- The expected prize of one ticket is $\frac{500+1000+1000+500}{1000}=3$ thousand dinars.
Result
| Answer | |
|---|---|
| Probability of at least 20 thousand | $0.001+0.010+0.050=0.061$ |