Question 1202001: An ornithologist wants to catch a certain pair of birds - a female and a male. The birds are caught one by one until a pair is formed. It is known that in such a bird population, females make up P% of the population. On average, how many birds will the ornithologist have to catch? (P = 33)
Answer by asinus(45)   (Show Source):
You can put this solution on YOUR website! Certainly, let's find the average number of birds the ornithologist needs to catch.
**1. Define Events**
* Let 'F' represent catching a female bird.
* Let 'M' represent catching a male bird.
**2. Calculate Probabilities**
* Probability of catching a female bird: P(F) = P/100 = 0.33
* Probability of catching a male bird: P(M) = 1 - P(F) = 1 - 0.33 = 0.67
**3. Determine Possible Scenarios**
* **Scenario 1: F, M**
* Catch a female first, then a male.
* Probability: P(F) * P(M) = 0.33 * 0.67 = 0.2211
* **Scenario 2: M, F**
* Catch a male first, then a female.
* Probability: P(M) * P(F) = 0.67 * 0.33 = 0.2211
**4. Calculate the Expected Number of Birds to Catch in Each Scenario**
* **Scenario 1:** 2 birds
* **Scenario 2:** 2 birds
**5. Calculate the Overall Expected Number of Birds**
* Expected number of birds = (Probability of Scenario 1 * Birds in Scenario 1) + (Probability of Scenario 2 * Birds in Scenario 2)
* Expected number of birds = (0.2211 * 2) + (0.2211 * 2) = 0.4422 + 0.4422 = 0.8844
**Therefore, on average, the ornithologist will need to catch approximately 0.8844 * 2 = 1.7688 birds.**
**Since it's not possible to catch a fraction of a bird, the ornithologist would need to catch an average of 2 birds to form a pair.**
**Note:**
* This calculation assumes that the population of birds is large enough that the probabilities of catching a female or male bird remain relatively constant throughout the process.
Let me know if you have any other questions!
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