To find the expected cash winnings when drawing one card at random from a jar containing four cards with values 1, 1, 2, and 16 dollars, we need to calculate the expected value. The expected value (EV) is a measure of the average outcome of a random event over a large number of trials. In this context, it's the average cash winnings per draw.
Here's a step-by-step approach:
1. List the Card Values:
The values of the cards are 1 dollar, 1 dollar, 2 dollars, and 16 dollars.
2. Determine the Probability of Drawing Each Card:
Since there are 4 cards and each card is equally likely to be drawn, the probability of drawing any one specific card is:
[tex]\[
P(\text{each card}) = \frac{1}{4}
\][/tex]
3. Calculate the Contribution of Each Card to the Expected Value:
Each card's contribution to the expected value is its value multiplied by the probability of drawing that card.
For the card with 1 dollar:
[tex]\[
\text{Contribution} = 1 \times \frac{1}{4} = \frac{1}{4}
\][/tex]
For the second card with 1 dollar (since it's identical to the first):
[tex]\[
\text{Contribution} = 1 \times \frac{1}{4} = \frac{1}{4}
\][/tex]
For the card with 2 dollars:
[tex]\[
\text{Contribution} = 2 \times \frac{1}{4} = \frac{2}{4} = \frac{1}{2}
\][/tex]
For the card with 16 dollars:
[tex]\[
\text{Contribution} = 16 \times \frac{1}{4} = 4
\][/tex]
4. Sum All Contributions:
[tex]\[
\text{Expected Value} = \frac{1}{4} + \frac{1}{4} + \frac{1}{2} + 4
\][/tex]
Simplify the sum:
[tex]\[
\text{Expected Value} = \frac{1 + 1 + 2 + 16}{4} = \frac{20}{4} = 5
\][/tex]
So, the person's expected cash winnings are:
[tex]\[
\boxed{5} \text{ dollars}
\][/tex]
