Question

Math

Posted 4 months ago

```
Martha has planted 4 root vegetables to make chili to enter into The Southwest Chili Cook-off.
Each vegetable that she plants has a $50 \%$ chance of successfully growing independent of the other vegetables. For each unsuccessful vegetable, she will have to use canned vegetables, which will be much less tasty, and result in less prize winnings for Martha.
The table below shows how much Martha will win in the cook-off based on the number of successfully grown vegetables.
What is Martha's expected winnings from The Southwest Chili Cook-off? Round your answer to the nearest cent. $\$$ $\square$
\begin{tabular}{lrrrrr}
\hline Successful vegetables & 0 & 1 & 2 & 3 & 4 \\
Winnings & $\$ 0$ & $\$ 10$ & $\$ 50$ & $\$ 75$ & $\$ 150$
\end{tabular}
```

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Answer from Sia

Posted 4 months ago

Solution by Steps

step 1

Calculate the probability-weighted winnings for 0 successful vegetables: $0.5^0 \times (1 - 0.5)^4 \times \$0$

step 2

Calculate the probability-weighted winnings for 1 successful vegetable: $4 \times 0.5^1 \times (1 - 0.5)^3 \times \$10$

step 3

Calculate the probability-weighted winnings for 2 successful vegetables: $6 \times 0.5^2 \times (1 - 0.5)^2 \times \$50$

step 4

Calculate the probability-weighted winnings for 3 successful vegetables: $4 \times 0.5^3 \times (1 - 0.5)^1 \times \$75$

step 5

Calculate the probability-weighted winnings for 4 successful vegetables: $0.5^4 \times (1 - 0.5)^0 \times \$150$

step 6

Add the results from steps 1 to 5 to find Martha's expected winnings

Answer

Martha's expected winnings are \$49.375, which rounds to \$49.38.

Key Concept

Expected Value in Probability

Explanation

The expected value is calculated by multiplying each outcome by its probability and summing these products. It represents the average amount one can expect to win per bet if the bet is repeated many times.

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