Math

QuestionA jar has 18 red and 40 blue marbles. If you pick 2 marbles at once:
a) Are the events dependent or independent? b) Why? c) What is the probability both are red?

Studdy Solution

STEP 1

Assumptions1. The jar contains18 red marbles and40 blue marbles. . Two marbles are drawn at the same time.
3. The marbles are drawn without replacement.

STEP 2

To determine whether the events are dependent or independent, we need to understand the definitions of these terms.Independent events are those where the outcome of one event does not affect the outcome of the other. Dependent events, on the other hand, are those where the outcome of one event does affect the outcome of the other.

STEP 3

In this case, because we are drawing two marbles at the same time without replacement, the outcome of the first draw (the color of the first marble) does affect the outcome of the second draw (the color of the second marble).Therefore, the events are dependent.

STEP 4

To find the probability that both marbles are red, we first need to find the total number of marbles in the jar.
Totalmarbles=Redmarbles+BluemarblesTotal\, marbles = Red\, marbles + Blue\, marbles

STEP 5

Plug in the given values for the number of red and blue marbles to calculate the total number of marbles.
Totalmarbles=18+40Total\, marbles =18 +40

STEP 6

Calculate the total number of marbles in the jar.
Totalmarbles=18+40=58Total\, marbles =18 +40 =58

STEP 7

The probability that the first marble drawn is red is the number of red marbles divided by the total number of marbles.
(Redfirst)=RedmarblesTotalmarbles(Red\, first) = \frac{Red\, marbles}{Total\, marbles}

STEP 8

Plug in the given values for the number of red marbles and the total number of marbles to calculate the probability that the first marble drawn is red.
(Redfirst)=1858(Red\, first) = \frac{18}{58}

STEP 9

After the first marble is drawn, there is one less red marble and one less total marble. The probability that the second marble drawn is red is the number of remaining red marbles divided by the number of remaining total marbles.
(Redsecond)=RedmarblesTotalmarbles(Red\, second) = \frac{Red\, marbles -}{Total\, marbles -}

STEP 10

Plug in the given values for the number of red marbles and the total number of marbles to calculate the probability that the second marble drawn is red.
(Redsecond)=1858(Red\, second) = \frac{18 -}{58 -}

STEP 11

Calculate the probability that the second marble drawn is red.
(Redsecond)=1757(Red\, second) = \frac{17}{57}

STEP 12

The probability that both marbles drawn are red is the product of the probability that the first marble drawn is red and the probability that the second marble drawn is red.
(Bothred)=(Redfirst)times(Redsecond)(Both\, red) =(Red\, first) \\times(Red\, second)

STEP 13

Plug in the calculated probabilities for the first and second draws to calculate the probability that both marbles drawn are red.
(Bothred)=1858times1757(Both\, red) = \frac{18}{58} \\times \frac{17}{57}

STEP 14

Calculate the probability that both marbles drawn are red.
(Bothred)=1858times17570.097(Both\, red) = \frac{18}{58} \\times \frac{17}{57} \approx0.097The probability that both marbles drawn are red is approximately0.097.

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