To find the expected value of the number shown after adding one additional slip with the value \(2\) to the bag, we begin by determining the total number of slips and their values.
Initially, the bag contains:
- 8 slips with a \(7\)
- 2 slips with a \(9\)
Adding together the slips, the counts and values are as follows:
- Total slips before adding the \(2\): \(10 \) (8 slips with \(7\) and 2 slips with \(9\))
- Total value before adding the \(2\): \[ 8 \times 7 + 2 \times 9 = 56 + 18 = 74 \]
When we add one slip with the value \(2\), the counts now become:
- Total slips now: \(11\) (8 slips with \(7\), 2 slips with \(9\), and 1 slip with \(2\))
Now, the new expected value \(E\) can be calculated as: \[ E = \frac{\text{total value}}{\text{total number of slips}} \]
The total value after the addition of the new slip: \[ \text{New total value} = 74 + 2 = 76 \]
Thus, the expected value is: \[ E = \frac{76}{11} \]
Now we perform the division to simplify \(E\): \[ E \approx 6.9091 \]
To summarize, the expected value of the number shown when picking a slip from the bag after adding the new slip with the number \(2\) is: \[ \boxed{\frac{76}{11}} \]
This is approximately \(6.9091\), but the exact expected value is more useful in this context. Hence we conclude the expected value is \(\frac{76}{11}\).
4 answers