To determine the person's expected cash winnings from drawing one card at random, we need to use the formula for the expected value (E) of a discrete random variable. The formula is:
[tex]\[ E = \sum (p_i \cdot x_i) \][/tex]
where [tex]\( p_i \)[/tex] is the probability of each outcome and [tex]\( x_i \)[/tex] is the value of each outcome.
Given the probabilities and the cash values:
- The probability of drawing a card worth [tex]$1 is 0.5.
- The probability of drawing a card worth $[/tex]5 is 0.4.
- The probability of drawing a card worth [tex]$20 is 0.1.
We will calculate the expected value step-by-step:
1. For the $[/tex]1 card:
[tex]\[
E_1 = p_1 \cdot x_1 = 0.5 \times 1 = 0.5
\][/tex]
2. For the [tex]$5 card:
\[
E_2 = p_2 \cdot x_2 = 0.4 \times 5 = 2.0
\]
3. For the $[/tex]20 card:
[tex]\[
E_3 = p_3 \cdot x_3 = 0.1 \times 20 = 2.0
\][/tex]
Now, add these individual expected values together to get the total expected value:
[tex]\[
E = E_1 + E_2 + E_3 = 0.5 + 2.0 + 2.0 = 4.5
\][/tex]
So, the person's expected cash winnings are [tex]\( \$ 4.5 \)[/tex].
