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A class has 10 students with ages 3, 3, 3, 3, 3, 4, 4, 4, 4, 5. If a student is chosen randomly, the probability of the student being age 3 is [tex]\frac{5}{10}[/tex] or 0.5, age 4 is [tex]\frac{4}{10}[/tex] or 0.4, and age 5 is [tex]\frac{1}{10}[/tex] or 0.1. What is the expected age of a randomly chosen student?



Answer :

GinnyAnswer
To find the expected age of a randomly chosen student from the class, we need to use the concept of expectation in probability. The expected value (or expectation) of a discrete random variable is calculated as the sum of the possible values each multiplied by their respective probabilities.

Here are the steps to find the expected age:

1. Identify the possible ages and their probabilities:

- Age 3: Probability = 0.5
- Age 4: Probability = 0.4
- Age 5: Probability = 0.1

2. Calculate the expected age using the formula for the expected value:

[tex]\[ \text{Expected Age} = (3 \times 0.5) + (4 \times 0.4) + (5 \times 0.1) \][/tex]

3. Compute each term:

- [tex]\( 3 \times 0.5 = 1.5 \)[/tex]
- [tex]\( 4 \times 0.4 = 1.6 \)[/tex]
- [tex]\( 5 \times 0.1 = 0.5 \)[/tex]

4. Add the results of these computations:

[tex]\[ \text{Expected Age} = 1.5 + 1.6 + 0.5 \][/tex]

5. Sum up the terms to find the final expected age:

[tex]\[ \text{Expected Age} = 3.6 \][/tex]

Therefore, the expected age of a randomly chosen student from the class is [tex]\( \boxed{3.6} \)[/tex] years.

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