The problem in the image asks for the expected value $E(X)$, which is computed as the weighted sum of all possible outcomes using their probabilities. The formula for expected value is:

$E(X) = \sum [x \cdot P(x)]$

Information given:

• $P(0) = 0.19$
• $P(1) = 0.27$
• $P(2) = 0.21$
• $P(3) = 0.18$
• $P(4) = 0.15$

Steps to Calculate:

1. Multiply each number of offspring ($x$) by its probability ($P(x)$).
2. Add these products together.

Calculation:

$E(X) = (0 \cdot 0.19) + (1 \cdot 0.27) + (2 \cdot 0.21) + (3 \cdot 0.18) + (4 \cdot 0.15)$

Breaking it down:

• $0 \cdot 0.19 = 0$
• $1 \cdot 0.27 = 0.27$
• $2 \cdot 0.21 = 0.42$
• $3 \cdot 0.18 = 0.54$
• $4 \cdot 0.15 = 0.60$

Adding these together: $E(X) = 0 + 0.27 + 0.42 + 0.54 + 0.60 = 1.83$

Final Answer:

$E(X) = 1.83$

Let me know if you’d like a more detailed explanation!

Follow-up Questions:

1. What is the purpose of calculating the expected value in probability distributions?
2. Can $E(X)$ ever be a value not listed among the possible outcomes? Why?
3. How do we calculate $E(X)$ if the probabilities do not add up to 1?
4. Why does $E(X)$ sometimes not match the most probable outcome?
5. What is the significance of $E(X)$ in real-world scenarios involving offspring?

Tip: Always ensure that the probabilities sum up to 1 before calculating $E(X)$.