To find the expected value of the number shown when drawing a slip of paper from the bag, we can use the formula for expectation, which is defined as:
\[ E[X] = \sum (x_i \cdot P(x_i)) \]
where \( x_i \) is a possible outcome and \( P(x_i) \) is the probability of that outcome.
In our case, we have two different outcomes from the slips of paper: the numbers \( 7 \) and \( 9 \).
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Calculate the probabilities of each outcome:
- There are \( 8 \) slips with a \( 7 \).
- There are \( 2 \) slips with a \( 9 \).
- The total number of slips is \( 10 \).
The probabilities for each outcome are as follows:
- \( P(7) = \frac{8}{10} = 0.8 \)
- \( P(9) = \frac{2}{10} = 0.2 \)
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Calculate the expected value: Now we can compute the expected value using the probabilities and the outcomes:
\[ E[X] = (7 \cdot P(7)) + (9 \cdot P(9)) \]
Substituting the values:
\[ E[X] = (7 \cdot 0.8) + (9 \cdot 0.2) \]
Calculating each term:
- \( 7 \cdot 0.8 = 5.6 \)
- \( 9 \cdot 0.2 = 1.8 \)
Now we add these two results:
\[ E[X] = 5.6 + 1.8 = 7.4 \]
Therefore, the expected value of the number shown when we draw a single slip of paper is
\[ \boxed{7.4} \]
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