Answer :
Hello! I can help you with that.
To find the expected number of physicians in the delegation, we first need to calculate the probabilities of selecting different numbers of physicians.
1. Calculate the total number of ways to select a delegation of 3 from 4 physicians and 6 surgeons using combinations:
Total ways = 10 choose 3 = 10! / (3! * 7!) = 120 ways.
2. Calculate the number of ways to select 0, 1, 2, and 3 physicians:
- 0 physicians: Choose 3 surgeons from 6 surgeons = 6 choose 3 = 20 ways.
- 1 physician: Choose 1 physician from 4 physicians and 2 surgeons from 6 surgeons = 4 choose 1 * 6 choose 2 = 60 ways.
- 2 physicians: Choose 2 physicians from 4 physicians and 1 surgeon from 6 surgeons = 4 choose 2 * 6 choose 1 = 36 ways.
- 3 physicians: Choose 3 physicians from 4 physicians = 4 choose 3 = 4 ways.
3. Calculate the probability of selecting each scenario:
- P(0 physicians) = 20 / 120 = 1/6
- P(1 physician) = 60 / 120 = 1/2
- P(2 physicians) = 36 / 120 = 3/10
- P(3 physicians) = 4 / 120 = 1/30
4. Finally, calculate the expected number of physicians:
Expected number = 0 * P(0 physicians) + 1 * P(1 physician) + 2 * P(2 physicians) + 3 * P(3 physicians)
Expected number = 0 * 1/6 + 1 * 1/2 + 2 * 3/10 + 3 * 1/30
Expected number = 0 + 0.5 + 0.6 + 0.1
Expected number = 1.2
Therefore, the expected number of physicians in the delegation is 1.2.