Problem 3) A box contains four 40-W, five 60-W, and 6 75-W light bulbs. If bulbs are selected one by one in random order, what is the probability that at least two bulbs should be selected to obtain one that is rated 75-W?
Problem 3) A box contains four 40-W, five 60-W, and 6 75-W light bulbs. If bulbs are selected one by one in random order, what is the probability that at least two bulbs should be selected to obtain one that is rated 75-W?
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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![### Problem 3)
A box contains four 40-W, five 60-W, and six 75-W light bulbs. If bulbs are selected one by one in random order, what is the probability that at least two bulbs should be selected to obtain one that is rated 75-W?
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To solve this problem, we need to determine the probability of selecting at least two bulbs in order to get one that is 75-W rated. The key is to calculate the complementary probability that the first selected bulb is not 75-W and then subtract that from 1.
Let's define:
- \( P(\text{first bulb not 75-W}) \) = Probability that the first selected bulb is not 75-W.
There are a total of \( 4 (40-W) + 5 (60-W) + 6 (75-W) = 15 \) light bulbs.
The probability that the first selected bulb is not 75-W is:
\[ P(\text{first bulb not 75-W}) = \frac{4 + 5}{15} = \frac{9}{15} = \frac{3}{5} \]
The probability that the first selected bulb is a 75-W bulb is:
\[ P(\text{first bulb is 75-W}) = \frac{6}{15} = \frac{2}{5} \]
Now, we are interested in the complementary event that we do not get a 75-W bulb on the first draw. So the probability of getting at least one 75-W bulb in two draws (at least two bulbs need to be selected) is given by:
\[ P(\text{at least two bulbs selected to get 75-W}) = 1 - P(\text{first bulb is 75-W}) = 1 - \frac{2}{5} = \frac{3}{5} \]
Therefore, the probability that at least two bulbs should be selected to obtain one that is rated 75-W is \( \frac{3}{5} \) or 60%.
This approach illustrates the basic principles of probability relevant to randomly choosing objects and calculating the likelihood of specific outcomes in a sequence of draws.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4fcf4b0b-d82c-43e8-86bb-da9864dd2a82%2F44c66295-e035-4b48-9999-af8265bf466c%2Ffqzgakd_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 3)
A box contains four 40-W, five 60-W, and six 75-W light bulbs. If bulbs are selected one by one in random order, what is the probability that at least two bulbs should be selected to obtain one that is rated 75-W?
---
To solve this problem, we need to determine the probability of selecting at least two bulbs in order to get one that is 75-W rated. The key is to calculate the complementary probability that the first selected bulb is not 75-W and then subtract that from 1.
Let's define:
- \( P(\text{first bulb not 75-W}) \) = Probability that the first selected bulb is not 75-W.
There are a total of \( 4 (40-W) + 5 (60-W) + 6 (75-W) = 15 \) light bulbs.
The probability that the first selected bulb is not 75-W is:
\[ P(\text{first bulb not 75-W}) = \frac{4 + 5}{15} = \frac{9}{15} = \frac{3}{5} \]
The probability that the first selected bulb is a 75-W bulb is:
\[ P(\text{first bulb is 75-W}) = \frac{6}{15} = \frac{2}{5} \]
Now, we are interested in the complementary event that we do not get a 75-W bulb on the first draw. So the probability of getting at least one 75-W bulb in two draws (at least two bulbs need to be selected) is given by:
\[ P(\text{at least two bulbs selected to get 75-W}) = 1 - P(\text{first bulb is 75-W}) = 1 - \frac{2}{5} = \frac{3}{5} \]
Therefore, the probability that at least two bulbs should be selected to obtain one that is rated 75-W is \( \frac{3}{5} \) or 60%.
This approach illustrates the basic principles of probability relevant to randomly choosing objects and calculating the likelihood of specific outcomes in a sequence of draws.
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