Question

1. a month of the year is selected at random. what is the probability that it has 31 days or starts with the letter j? a. 1/6 c. 7/12 b. 3/4 d. 2/3 7. a card is selected at random from a standard deck of 52 playing cards. what is the probability that it is not a face card? a. 3/13 c. 4/13 b. 10/13 d. 9/13 8. a die is rolled, then a letter in the word statistics is randomly selected. what is the probability of rolling at least a three, then selecting the letter s? a. 1/5 c. 3/20 b. 1/4 d. 29/30 use the following situation for #9 - 10 a basket contains 8 green apples, 11 red apples, 5 nectarines, and 12 oranges. mark randomly chooses a piece of fruit, eats it, then selects another. 9. p(a green apple then a nectarine) a. 23/63 c. 13/36 b. 5/162 d. 2/63 10. p(two oranges) a. 1/9 c. 11/105 b. 4/35 d. 68/105

Explanation:

Question 6

Step1: Identify months with 31 days or starting with J

Months with 31 days: January, March, May, July, August, October, December (7 months).
Months starting with J: January, June, July (3 months).
Overlap (January, July): 2 months.
Using inclusion - exclusion: \( n(A\cup B)=n(A)+n(B)-n(A\cap B)=7 + 3-2=8 \)? Wait, no, wait. Wait, January: 31 days and starts with J; July: 31 days and starts with J; June: starts with J (30 days). So months with 31 days: Jan, Mar, May, Jul, Aug, Oct, Dec (7). Months starting with J: Jan, Jun, Jul (3). So \( A\cup B \): Jan, Mar, May, Jul, Aug, Oct, Dec, Jun. Wait, that's 8? Wait, no, let's list all:
Months: 12 total.
Months with 31 days: Jan (31, J), Mar (31), May (31), Jul (31, J), Aug (31), Oct (31), Dec (31) → 7.
Months starting with J: Jan (J), Jun (J), Jul (J) → 3.
The union: Jan, Mar, May, Jul, Aug, Oct, Dec, Jun → 8? Wait, no, 7 + 3 - 2 (Jan and Jul are in both) = 8. Wait, but the answer options: let's recalculate. Wait, maybe I made a mistake. Wait, January: 31 days, starts with J; June: starts with J (30 days); July: 31 days, starts with J. So months with 31 days or starts with J: Jan, Mar, May, Jul, Aug, Oct, Dec, Jun. That's 8 months? Wait, but 8/12 = 2/3. Oh! Wait, 8 months? Wait, 7 (31 - day) + 3 (J - start) - 2 (overlap: Jan, Jul) = 8. 8/12 = 2/3.

Step2: Calculate probability

Probability \( P=\frac{8}{12}=\frac{2}{3} \)

Step1: Number of face cards and non - face cards

In a standard deck, face cards: Jack, Queen, King. 3 face cards per suit, 4 suits. So \( 3\times4 = 12 \) face cards.
Non - face cards: \( 52-12 = 40 \).

Step2: Calculate probability

Probability of not a face card: \( P=\frac{40}{52}=\frac{10}{13} \)

Step1: Probability of rolling at least 3 on a die

A die has 6 faces. Rolling at least 3 means 3,4,5,6. So number of favorable outcomes = 4. Probability \( P_1=\frac{4}{6}=\frac{2}{3} \).

Step2: Probability of selecting S from "STATISTICS"

The word "STATISTICS" has letters: S, T, A, T, I, S, T, I, C, S. So total letters: 10. Number of S: 3. Probability \( P_2=\frac{3}{10} \).

Step3: Probability of both events (independent)

Since the die roll and letter selection are independent, \( P = P_1\times P_2=\frac{2}{3}\times\frac{3}{10}=\frac{1}{5} \)? Wait, no, wait. Wait, "STATISTICS": let's count again. S, T, A, T, I, S, T, I, C, S. So letters: S (3), T (3), A (1), I (2), C (1). Total: 3 + 3+1 + 2+1 = 10. So \( P_2=\frac{3}{10} \). \( P_1=\frac{4}{6}=\frac{2}{3} \). Then \( P=\frac{2}{3}\times\frac{3}{10}=\frac{1}{5} \). Wait, but option A is 1/5. Wait, but let's check the die roll: "at least a three" means 3,4,5,6: 4 outcomes out of 6, so \( \frac{4}{6}=\frac{2}{3} \). Letter S: 3 out of 10. So \( \frac{2}{3}\times\frac{3}{10}=\frac{1}{5} \).

Answer:

D. 2/3

Question 7