jeff has 8 red marbles, 6 blue marbles, and 4 green marbles that are the same size and shape. he puts the marbles into a bag, mixes the marbles, and randomly picks one marble. what is the probability that the marble will be blue?
The Correct Answer and Explanation is:
To find the probability that the marble Jeff picks is blue, we use the basic formula for probability: Probability=Number of favorable outcomesTotal number of possible outcomes\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}
Step 1: Count the total number of marbles
Jeff has:
- 8 red marbles
- 6 blue marbles
- 4 green marbles
So, the total number of marbles is: 8+6+4=188 + 6 + 4 = 18
Step 2: Identify the number of favorable outcomes
We are looking for the probability of picking a blue marble. There are 6 blue marbles.
Step 3: Calculate the probability
Probability of blue=618=13\text{Probability of blue} = \frac{6}{18} = \frac{1}{3}
Final Answer:
13\boxed{\frac{1}{3}}
300-Word Explanation:
Probability is a branch of mathematics that deals with the likelihood of different outcomes. In this problem, Jeff has a total of 18 marbles—8 red, 6 blue, and 4 green—all of which are indistinguishable in size and shape. When he randomly selects one marble, each marble has an equal chance of being picked because the selection is random and all marbles are equally likely to be chosen.
The event of interest here is that Jeff picks a blue marble. There are 6 blue marbles among the 18 total marbles. To find the probability of this event, we divide the number of favorable outcomes (blue marbles) by the total number of outcomes (all marbles), which gives us the formula: Number of blue marblesTotal number of marbles=618\frac{\text{Number of blue marbles}}{\text{Total number of marbles}} = \frac{6}{18}
This simplifies to 13\frac{1}{3}, meaning that out of every three times Jeff picks a marble, we expect (on average) one of those picks to be a blue marble, assuming he replaces the marble each time.
Understanding this concept helps in real-life situations involving risk, odds, or chance—like drawing a card from a deck or rolling dice. It also lays the foundation for more complex probability topics such as dependent events, combinations, and permutations. In summary, the probability that Jeff picks a blue marble is 13\boxed{\frac{1}{3}}.