SOLUTION: Bottle 1 consists of 4 red balls and 6 blue balls. Bottle 2 consists of some unknown number of red balls and 12 blue balls. One ball is drawn from each bottle. The probability th

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Question 945409: Bottle 1 consists of 4 red balls and 6 blue balls. Bottle 2 consists of some unknown number of red balls and 12 blue balls. One ball is drawn from each bottle. The probability that both balls are red or both balls are blue is 0.56. Find the number of red balls in bottle 2
Answer by mathmate(429) About Me  (Show Source):
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Given:
Two bottles containing red and blue balls.
Bottle 1: 4R+6B
Bottle 2: xR+12B
One ball is drawn from each bottle.
P(RR∪BB)=0.56 (i.e. probability of drawing both red or both blue)
Find x (number of red balls in bottle 2).

Solution:
Let
R=event of drawing a red ball
B=event of drawing a blue ball
Since events of drawing from different bottles are independent, we can multiply the individual probabilities, i.e.
P(RR)=P(R) from bottle 1 * P(R) from bottle 2, and similarly for P(BB)

P(RR)=%284%2F%284%2B6%29%29%2A%28x%2F%28x%2B12%29%29=0.4x%2F%28x%2B12%29
P(BB)=%286%2F%284%2B6%29%29%2A%2812%2F%28x%2B12%29%29=7.2%2F%28x%2B12%29
Since the events RR and BB are mutually exclusive, probability of the union is the sum of the individual probabilities.
P(RR∪BB)=0.4x%2F%28x%2B12%29%2B7.2%2F%28x%2B12%29
which is given to be 0.56
So we have an equation in x from which we can solve for x:
0.4x%2F%28x%2B12%29%2B7.2%2F%28x%2B12%29+=+0.56
0.4x%2B7.2=0.56%28x%2B12%29
7.2-6.72=0.16x
x=0.48%2F0.16=3

Answer:
There are 3 red balls in the second bottle.