Question

A certain jar contains 100 jelly beans: 50 white, 30 green, 10 yellow, 5 red, 4 purple, and 1 black. If a jelly bean is to be chosen at random, what is the probability that the jelly bean will be neither purple nor red?

• A. 0.09
• B. 0.11
• C. 0.55
• D. 0.91
• E. 0.96

Hint:

For "neither/nor" probability questions, using the complement rule (the indirect method) is often faster. It's usually easier to count the items you want to exclude and subtract their probability from 1.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This is a probability problem. We need to find the probability of a specific event occurring when an item is chosen at random from a set. The event is that the chosen jelly bean is "neither purple nor red."
Step 2: Key Formula or Approach:
The probability of an event is calculated as: \[ P(\text{Event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \] There are two ways to solve this: 1. Direct Method: Count the number of jelly beans that are "neither purple nor red" (the favorable outcomes) and divide by the total number of jelly beans. 2. Indirect Method (Complement): Calculate the probability of the opposite event (the jelly bean IS purple or red) and subtract this from 1. \( P(\text{not } E) = 1 - P(E) \).
Step 3: Detailed Explanation:
Total number of jelly beans = 100. Method 1: Direct Method "Favorable outcomes" are the jelly beans that are not purple and not red. These are the white, green, yellow, and black jelly beans. Number of white = 50 Number of green = 30 Number of yellow = 10 Number of black = 1 Total number of favorable outcomes = \( 50 + 30 + 10 + 1 = 91 \). Now, calculate the probability: \[ P(\text{neither purple nor red}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{91}{100} = 0.91 \] Method 2: Indirect Method (Complement) First, find the number of jelly beans that ARE purple or red. Number of purple = 4 Number of red = 5 Total number of "unfavorable" outcomes = \( 4 + 5 = 9 \). The probability of picking a purple or red jelly bean is: \[ P(\text{purple or red}) = \frac{9}{100} = 0.09 \] The probability of the jelly bean being NEITHER purple nor red is the complement of this event. \[ P(\text{neither purple nor red}) = 1 - P(\text{purple or red}) = 1 - 0.09 = 0.91 \] Step 4: Final Answer:
Both methods show that the probability is 0.91.