QuestionA bag has 4 red and 7 white balls. If a ball is drawn and replaced with the other color, what is the probability the second is red?
Studdy Solution
STEP 1
Assumptions1. The bag initially contains4 red balls and7 white balls.
. After the first ball is drawn, a ball of the other color is put into the bag.
3. We are interested in the probability that the second ball drawn is red.
STEP 2
First, we need to consider the two possible scenarios for the first draw either a red ball or a white ball is drawn.
STEP 3
If a red ball is drawn first, then a white ball is added to the bag. This changes the composition of the bag to3 red balls and8 white balls.
STEP 4
The probability of drawing a red ball on the second draw, given that a red ball was drawn first, is therefore
STEP 5
Calculate the probability.
STEP 6
If a white ball is drawn first, then a red ball is added to the bag. This changes the composition of the bag to5 red balls and6 white balls.
STEP 7
The probability of drawing a red ball on the second draw, given that a white ball was drawn first, is therefore
STEP 8
Calculate the probability.
STEP 9
Now, we need to consider the probabilities of each of the first draw scenarios occurring. The probability of drawing a red ball first is
STEP 10
Calculate the probability.
STEP 11
The probability of drawing a white ball first is
STEP 12
Calculate the probability.
STEP 13
Finally, we can calculate the total probability of drawing a red ball on the second draw by summing the probabilities of each scenario, each weighted by the probability of that scenario occurring.
STEP 14
Plug in the values for the probabilities.
STEP 15
Calculate the total probability.
The probability that the second ball drawn is red is .
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