Question

working with mutually exclusive events
each letter in the word statistics is written on an index card, then shuffled. one card is drawn at random.
let a = the card drawn is a vowel. let b = the card drawn is the letter t.
there are 10 letters are in the word statistics. 3 of which are vowels.

• what is p(a)?
• what is p(b)?
• how many of the 10 letters are vowels or the letter t?
• what is p(a or b)?
• what is p(a) + p(b)?

Explanation:

1. What is \( P(A) \)?

Step 1: Identify total and favorable outcomes

Total letters in "STATISTICS": 10. Vowels (A, I, I): 3. So \( P(A)=\frac{\text{Number of vowels}}{\text{Total letters}} \).

Step 2: Calculate \( P(A) \)

\( P(A)=\frac{3}{10} = 0.3 \)

Step 1: Count vowels, T, and overlaps

Vowels: 3 (A, I, I). T's: 3 (from "STATISTICS": T, T, T). No overlap (vowels are A, I; T is consonant). So total = vowels + T's.

Step 2: Calculate total

\( 3 + 3 = 6 \)

Step 1: Identify total and favorable for B

Total letters: 10. T's: 3. So \( P(B)=\frac{\text{Number of T's}}{\text{Total letters}} \).

Step 2: Calculate \( P(B) \)

\( P(B)=\frac{3}{10} = 0.3 \)

Answer:

\( \frac{3}{10} \) (or 0.3)

2. How many of the 10 letters are vowels or the letter T?