A lottery contains 100 tickets. It costs exactly one dollar to play. (And you don’t get your dollar back if you win!) There are 10 prizes: five 1dollar prizes, three 5 dollar prizes and two 20 dollar prizes. Let X be the amount won in the lottery. The distribution table for X, the net amount of money won is written as:
X | -1 | 1 | 5 | 20
P(X) | 0.85 | 0.05 | 0.03 | 0.02
X | -99 | -95 | -80 | 1 | 5 | 10 | 20 |
p(X) | 0.05 | 0.03 | 0.02 | 0.55 | 0.03 | 0.02 | 0.3 |
The table represents the distribution of the net amount of money won, denoted by X, in the lottery. For example, the probability of winning $1 is 0.55, while the probability of winning $20 is 0.3.
To calculate the distribution table for the amount of money won in the lottery (X), we need to consider the probabilities of winning each prize and the corresponding net amount won for each prize.
Given the information provided, let's calculate the probabilities and the net amount won for each prize:
1. Five 1-dollar prizes:
- Probability of winning each 1-dollar prize: 5/100 (since there are 5 1-dollar prizes out of 100 tickets)
- Net amount won for each 1-dollar prize: 1 dollar
- Total net amount won for all 1-dollar prizes: 5 dollars
2. Three 5-dollar prizes:
- Probability of winning each 5-dollar prize: 3/100 (since there are 3 5-dollar prizes out of 100 tickets)
- Net amount won for each 5-dollar prize: 5 dollars
- Total net amount won for all 5-dollar prizes: 15 dollars
3. Two 20-dollar prizes:
- Probability of winning each 20-dollar prize: 2/100 (since there are 2 20-dollar prizes out of 100 tickets)
- Net amount won for each 20-dollar prize: 20 dollars
- Total net amount won for all 20-dollar prizes: 40 dollars
To construct the distribution table, we calculate the total net amount won for each possible outcome of X (net amount of money won):
- If X is 1 dollar, the probability of winning 1 dollar is 5/100. Therefore, the total net amount won for this outcome is 5 dollars.
- If X is 5 dollars, the probability of winning 5 dollars is 3/100. Therefore, the total net amount won for this outcome is 15 dollars.
- If X is 20 dollars, the probability of winning 20 dollars is 2/100. Therefore, the total net amount won for this outcome is 40 dollars.
- If X is 0 dollars (no prize is won), the remaining probability is 1 - (5/100 + 3/100 + 2/100) = 90/100. Therefore, the total net amount won for this outcome is 0 dollars.
Considering these outcomes, the distribution table for X, the net amount of money won, is as follows:
X | Probability | Net Amount Won
----------------
1 | 5/100 | 5 dollars
5 | 3/100 | 15 dollars
20 | 2/100 | 40 dollars
0 | 90/100 | 0 dollars