Ellen has a bag with $3$ red marbles and $2$ blue marbles in it. She is going to randomly draw a marble from the bag $300$ times, putting the marble back in the bag after each draw. $\newline$ What is the best prediction for the number of times that Ellen will draw a blue marble? $\newline$ Choose $1$ answer: $\newline$ (A) Exactly $120$ times $\newline$ (B) Close to $120$ times but probably not exactly $120$ times $\newline$ (C) Exactly $150$ times $\newline$ (D) Close to $150$ times but probably not exactly $150$ times

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Math Problems

Algebra 2

Find probabilities using combinations and permutations

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Q. Ellen has a bag with

3

red marbles and

2

blue marbles in it. She is going to randomly draw a marble from the bag

300

times, putting the marble back in the bag after each draw.

\newline

What is the best prediction for the number of times that Ellen will draw a blue marble?

\newline

Choose

1

answer:

\newline

(A) Exactly

120

times

\newline

(B) Close to

120

times but probably not exactly

120

times

\newline

150

times

\newline

(D) Close to

150

times but probably not exactly

150

times

Determine Probability of Blue Marble: First, we need to determine the probability of drawing a blue marble in a single draw. Since there are $3$ red marbles and $2$ blue marbles, the total number of marbles is $3 + 2 = 5$ . The probability of drawing a blue marble is the number of blue marbles divided by the total number of marbles.
Calculate Probability of Drawing Blue Marble: Now, let's calculate the probability of drawing a blue marble. Probability of drawing a blue marble $=$ Number of blue marbles $/$ Total number of marbles $=$ $\frac{2}{5}$ .
Predict Number of Blue Marble Draws: Next, we will use this probability to predict the number of times a blue marble will be drawn out of $300$ draws. $\newline$ Since each draw is independent and the marble is replaced each time, we can multiply the probability of drawing a blue marble by the total number of draws to get the expected number of times a blue marble will be drawn.
Perform Calculation for Expected Draws: Let's perform the calculation for the expected number of blue marble draws. Expected number of blue marble draws = Probability of drawing a blue marble $\times$ Total number of draws = $(\frac{2}{5}) \times 300$ .
Compute Expected Number of Blue Marble Draws: Now, we will compute the expected number of blue marble draws. Expected number of blue marble draws = $(\frac{2}{5}) \times 300 = 120$ .
Interpret Result: Finally, we need to interpret the result. The calculation gives us the expected value, which is the best prediction for the average outcome over many trials. However, in any given set of $300$ draws, the actual number of times a blue marble is drawn is likely to be close to but not exactly $120$ times due to the randomness of each draw.