Question

1. darryl and danny buy a box of a dozen donuts. seven of the donuts are plain - glazed, 3 of the donuts are chocolate - covered and 2 of the donuts are vanilla - glazed. what is the probability of darryl choosing a plain - glazed donut, eating it, and then danny choosing another plain - glazed donut? a $\frac{7}{44}$ b $\frac{49}{144}$ c $\frac{7}{22}$ d $\frac{7}{24}$
2. for classroom awards, ms. blake has coupons in a bag that students can draw at random. in the bag, she has 6 coupons (that say \100 hall the assignment\) and 4 coupons that say \the homework\...

Explanation:

Step1: Determine total donuts initially

A dozen donuts is 12. So total donuts \( n = 12 \).

Step2: Determine number of plain - glazed donuts

Number of plain - glazed donuts \( p = 7 \).

Step3: Probability Daryl chooses plain - glazed

Probability \( P_1=\frac{\text{Number of plain - glazed}}{\text{Total number of donuts}}=\frac{7}{12}\)? Wait, no, wait. Wait, after Daryl eats one plain - glazed, then Danny chooses. Wait, let's re - read. Daryl and Danny buy a box of a dozen donuts. Seven of the donuts are plain - glazed, 3 are chocolate - covered, 2 are vanilla - glazed. What is the probability of Daryl choosing a plain - glazed donut, eating it, and then Danny choosing another plain - glazed donut?

First, total donuts \( N = 12 \), number of plain - glazed \( n_1=7 \).

Probability Daryl chooses plain - glazed: \( P(Daryl)=\frac{7}{12}\)? Wait, no, the options have denominators 44, 22, 144, 24. Wait, maybe I misread the number of donuts. Wait, maybe it's a box with more? Wait, maybe the problem is: Daryl and Danny buy a box of donuts. Let's check the options. Let's assume that the total number of donuts is 12 initially. Daryl picks a plain - glazed (7 out of 12), then after eating, the number of plain - glazed is 6 and total donuts is 11? No, the options don't match. Wait, maybe the total number of donuts is 24? Wait, maybe the problem is: Seven of the donuts are plain - glazed, 3 are chocolate - covered, 2 are vanilla - glazed, and maybe other types? Wait, 7 + 3+2 = 12. Wait, the options: A is \( \frac{7}{44} \), B is \( \frac{49}{144} \), C is \( \frac{7}{22} \), D is \( \frac{7}{24} \).

Wait, maybe the total number of donuts is 12, but when Daryl chooses, then Danny chooses, it's a two - step probability. Wait, no, let's re - express. Let's suppose that the total number of donuts is 12. Probability Daryl picks plain - glazed: \( \frac{7}{12} \). Then, after Daryl eats one plain - glazed, the number of plain - glazed is 6 and total donuts is 11. Then probability Danny picks plain - glazed is \( \frac{6}{11} \). Then the combined probability is \( \frac{7}{12}\times\frac{6}{11}=\frac{7\times6}{12\times11}=\frac{7}{22} \). Ah, that matches option C.

Step1: Probability Daryl picks plain - glazed

Total donuts \( T = 12 \), plain - glazed donuts \( P = 7 \). So probability \( P_1=\frac{7}{12} \).

Step2: Probability Danny picks plain - glazed after Daryl

After Daryl eats one plain - glazed, plain - glazed donuts left \( P'=7 - 1=6 \), total donuts left \( T'=12 - 1 = 11 \). So probability \( P_2=\frac{6}{11} \).

Step3: Combined probability

The combined probability is \( P = P_1\times P_2=\frac{7}{12}\times\frac{6}{11}=\frac{7\times6}{12\times11}=\frac{7}{22} \).

Answer:

C. \(\frac{7}{22}\)