Question

1. a bag of hard candy contains 4 pieces of butterscotch candy, 5 pieces of caramel candy, and 3 pieces of toffee candy. dan reaches into the bag and pulls out 2 pieces of candy. what is the probability that dan pulled out a caramel and a toffee? 1st choice 2nd choice $\frac{3}{11}$ b $\frac{4}{12}$ b $\frac{5}{11}$ c $\frac{3}{11}$ t $\frac{4}{11}$ b $\frac{5}{12}$ c $\frac{4}{11}$ c $\frac{3}{11}$ t $\frac{4}{11}$ b $\frac{3}{12}$ t $\frac{5}{11}$ c $\frac{2}{11}$ t $\frac{5}{44}$ $\frac{5}{22}$ $\frac{25}{1936}$

Explanation:

Step1: Calculate total number of candies

The total number of candies is $4 + 5+3=12$.

Step2: Calculate probability of two - case scenarios

There are two cases: first caramel then toffee and first toffee then caramel.

Case 1: First caramel then toffee

The probability of picking a caramel first is $\frac{5}{12}$. After picking a caramel, there are 11 candies left. The probability of picking a toffee next is $\frac{3}{11}$. So the probability of this case is $\frac{5}{12}\times\frac{3}{11}=\frac{15}{132}$.

Case 2: First toffee then caramel

The probability of picking a toffee first is $\frac{3}{12}$. After picking a toffee, there are 11 candies left. The probability of picking a caramel next is $\frac{5}{11}$. So the probability of this case is $\frac{3}{12}\times\frac{5}{11}=\frac{15}{132}$.

Step3: Calculate total probability

The total probability of getting a caramel and a toffee is $\frac{15}{132}+\frac{15}{132}=\frac{30}{132}=\frac{5}{22}$.

Answer:

$\frac{5}{22}$