Question

Players in a card game use the spinner below to determine how many cards they will pick up during their turn. The table below shows the probability distribution for the number of cards a player will pick up during one turn.
What is the expected value for the number of cards a player will pick up during one turn?
2
1
1
3
3
1
1
2
L
1
Number


of
Cards

Probability
1

0.5
2

0.3
3

0.2

Answer 1

To calculate the expected value (or mean) of the random variable, you can use the formula:

\[
E(X) = \sum (x_i \cdot P(x_i))
\]

where \(x_i\) represents the number of cards and \(P(x_i)\) represents the probability of that outcome.

From the information provided, we can set up the calculation as follows:

- For 1 card:
- \(x_1 = 1\)
- \(P(x_1) = 0.5\)

- For 2 cards:
- \(x_2 = 2\)
- \(P(x_2) = 0.3\)

- For 3 cards:
- \(x_3 = 3\)
- \(P(x_3) = 0.2\)

Now we can calculate the expected value:

\[
E(X) = (1 \cdot 0.5) + (2 \cdot 0.3) + (3 \cdot 0.2)
\]

Calculating each term:

- For 1 card: \(1 \cdot 0.5 = 0.5\)
- For 2 cards: \(2 \cdot 0.3 = 0.6\)
- For 3 cards: \(3 \cdot 0.2 = 0.6\)

Now sum these values:

\[
E(X) = 0.5 + 0.6 + 0.6 = 1.7
\]

Thus, the expected value for the number of cards a player will pick up during one turn is **1.7**.