To determine the person's expected cash winnings when they draw one card at random from a jar with four cards, we need to calculate the expected value of the card values. Here is a step-by-step guide to finding the expected value:
1. List the possible outcomes: The values of the cards in the jar are 1, 2, 4, and 5 dollars.
2. Assign probabilities to each outcome: Since each card is equally likely to be drawn, the probability for each card is [tex]\( \frac{1}{4} \)[/tex].
3. Calculate the expected value: The expected value (E) is found by multiplying each outcome by its probability and then summing these products.
[tex]\( E = (1 \cdot \frac{1}{4}) + (2 \cdot \frac{1}{4}) + (4 \cdot \frac{1}{4}) + (5 \cdot \frac{1}{4}) \)[/tex]
4. Compute the individual products:
- [tex]\( 1 \cdot \frac{1}{4} = \frac{1}{4} \)[/tex]
- [tex]\( 2 \cdot \frac{1}{4} = \frac{1}{4} \times 2 = \frac{2}{4} \)[/tex]
- [tex]\( 4 \cdot \frac{1}{4} = \frac{4}{4} = 1 \)[/tex]
- [tex]\( 5 \cdot \frac{1}{4} = \frac{5}{4} \)[/tex]
5. Sum these products:
[tex]\( E = \frac{1}{4} + \frac{2}{4} + 1 + \frac{5}{4} \)[/tex]
6. Combine the fractions to get the sum:
[tex]\( E = \frac{1}{4} + \frac{2}{4} + \frac{4}{4} + \frac{5}{4} = \frac{1+2+4+5}{4} = \frac{12}{4} = 3 \)[/tex]
Therefore, the person's expected cash winnings are [tex]$3.0$[/tex] dollars.
