Question

use your understanding of probability to answer the questions below. 7. harmony placed the letters in the word basketball into a bag. she wants to find the probability of drawing the letter b and then drawing a vowel. mark each statement below as true or false. correct any false statements. ____ a. if the first letter is replaced, the probability can be found by multiplying \\(\frac{2}{10}\\). __ b. if the first letter is replaced, the probability is \\(\frac{3}{50}\\). __ c. if the first letter is not replaced, the probability can be found by multiplying \\(\frac{2}{10}\cdot\frac{2}{10}\\). ____ d. if the first letter is not replaced, the probability is \\(\frac{1}{15}\\).

Explanation:

First, let's analyze the word "BASKETBALL". Let's count the letters: B, A, S, K, E, T, B, A, L, L. So total letters: 10.

Number of B's: 2.

Vowels in the word: A, E, A. So vowels: 3 (A, E, A). Wait, let's list all letters:

Letters: B, A, S, K, E, T, B, A, L, L. So vowels are A, E, A. So 3 vowels? Wait, A is a vowel, E is a vowel. Let's check:

Letters:

1. B (consonant)
2. A (vowel)
3. S (consonant)
4. K (consonant)
5. E (vowel)
6. T (consonant)
7. B (consonant)
8. A (vowel)
9. L (consonant)
10. L (consonant)

So vowels: A, E, A. So 3 vowels. Wait, but let's confirm: A, E, A. So three vowels.

Now, let's tackle each part:

Part a:

If the first letter is replaced, then the two events (drawing B, then drawing vowel) are independent.

Probability of drawing B: number of B's / total letters = 2/10.

Probability of drawing vowel: number of vowels / total letters = 3/10? Wait, wait, in the word BASKETBALL, let's list all letters again:

B, A, S, K, E, T, B, A, L, L. So letters: 10.

Vowels: A (position 2), E (position 5), A (position 8). So three vowels. So vowels count is 3. Wait, but maybe I made a mistake. Wait, A, E, A: that's three vowels. So probability of vowel is 3/10.

Wait, but the statement a says "multiplying 2/10 * 2/10". Wait, no, wait the original problem's a says: "If the first letter is replaced, the probability can be found by multiplying 2/10 * 2/10". Wait, but vowels are 3, so maybe I miscounted. Wait, let's check again.

Wait, BASKETBALL: let's spell it correctly. B-A-S-K-E-T-B-A-L-L. So letters: B, A, S, K, E, T, B, A, L, L. So that's 10 letters.

Vowels: A (1), E (1), A (1). So three vowels. So probability of vowel is 3/10. But the statement a says multiplying 2/10 * 2/10. Wait, maybe I made a mistake. Wait, maybe the vowels are A, E, A, but maybe the user considers A, E, A, and maybe another? Wait, no, the letters are 10. Wait, maybe the original problem has a typo, or maybe I miscounted. Wait, let's check again:

Wait, BASKETBALL: B, A, S, K, E, T, B, A, L, L. So vowels: A (2), E (1). So total vowels: 3 (A, A, E). So 3 vowels. So probability of vowel is 3/10.

But the statement a says "multiplying 2/10 * 2/10". Wait, that would be if vowels are 2, but they are 3. Wait, maybe the problem's original word is different? Wait, maybe I made a mistake. Wait, let's check the word BASKETBALL: B, A, S, K, E, T, B, A, L, L. So letters: 10. B's: 2. Vowels: A (positions 2 and 8), E (position 5). So that's three vowels (A, A, E). So 3 vowels. So probability of vowel is 3/10.

But the statement a says "multiplying 2/10 * 2/10". So that would be incorrect, because vowels are 3, not 2. Wait, maybe the problem has a different count. Wait, maybe the word is BASKETBALL with 10 letters, but maybe the vowels are A, E, A: 3. So statement a: if replaced, P(B) = 2/10, P(vowel) = 3/10. So multiplying 2/10 * 3/10 = 6/100 = 3/50. But the statement a says multiplying 2/10 * 2/10. So that's incorrect. Wait, maybe I miscounted vowels. Wait, maybe the word is BASKETBALL: let's check again. B, A, S, K, E, T, B, A, L, L. So vowels: A (2), E (1). So 3 vowels. So statement a is false? Wait, no, maybe the problem has a different approach. Wait, maybe the user made a typo, but let's proceed.

Wait, maybe the vowels are A, E, A: 3. So let's go through each part:

Part a:

If replaced, then P(B then vowel) = P(B) * P(vowel) (since independent). P(B) = 2/10, P(vowel) = 3/10. So 2/10 * 3/10 = 6/100 = 3/50. But the statement a says "multiplying 2/10 * 2/10". So that's incorrect. Wait, but maybe the vowels are 2? Wait, maybe I made a mistake. Wait, BASKETBALL: letters are B, A, S, K, E, T, B, A, L, L. So vowels: A, E, A. So 3. So statement a: says multiply 2/10 * 2/10. So that's wrong. So a is false? Wait, no, maybe the problem considers vowels as A, E: 2? Wait, no, A appears twice, E once. So three vowels.

Wait, maybe the original problem has a different word. Wait, maybe "BASKETBALL" has 10 letters, B:2, vowels: A, E, A: 3. So:

a. If replaced, P(B) = 2/10, P(vowel) = 3/10. So 2/10 * 3/10 = 6/100 = 3/50. The statement a says multiply 2/10 * 2/10. So that's incorrect. So a is false.

Part b:

If replaced, P = 2/10 * 3/10 = 6/100 = 3/50. So statement b says "the probability is 3/50". So that's correct. So b is true.

Part c:

If not replaced, then after drawing B, total letters become 9, and vowels: if we drew a B (consonant), then vowels are still 3. So P(B then vowel) = P(B) * P(vowel | B). P(B) = 2/10, P(vowel | B) = 3/9 (since one B is removed, total letters 9, vowels still 3). So 2/10 * 3/9 = 6/90 = 1/15. But statement c says multiply 2/10 * 2/10. So that's incorrect. So c is false.

Part d:

If not replaced, P(B then vowel) = 2/10 * 3/9 = 6/90 = 1/15. So statement d says "the probability is 1/15". So that's correct. So d is true.

Wait, but let's recheck:

Total letters: 10. B's: 2. Vowels: A, E, A (3). So:

- If replaced:
- P(B) = 2/10
- P(vowel) = 3/10
- So P(B then vowel) = 2/10 * 3/10 = 6/100 = 3/50. So statement b says 3/50, which is correct. So b is true.

- If not replaced:
- P(B) = 2/10
- After drawing B, total letters: 9, vowels still 3 (since B is consonant)
- P(vowel | B) = 3/9
- So P(B then vowel) = 2/10 * 3/9 = 6/90 = 1/15. So statement d says 1/15, which is correct. So d is true.

Now, let's check the statements:

a. If replaced, multiply 2/10 * 2/10. But vowels are 3, so P(vowel) is 3/10. So a is false.

b. If replaced, probability is 3/50. 2/10 * 3/10 = 6/100 = 3/50. So b is true.

c. If not replaced, multiply 2/10 * 2/10. But after drawing B, total letters 9, vowels 3, so P(vowel) is 3/9. So c is false.

d. If not replaced, probability is 1/15. 2/10 * 3/9 = 6/90 = 1/15. So d is true.

So:

a. False (should be 2/10 * 3/10)

b. True (2/10 * 3/10 = 6/100 = 3/50)

c. False (should be 2/10 * 3/9)

d. True (2/10 * 3/9 = 1/15)

Answer:

a. False (Correct: Multiply \( \frac{2}{10} \times \frac{3}{10} \))
b. True
c. False (Correct: Multiply \( \frac{2}{10} \times \frac{3}{9} \))
d. True