A bag contains 4 red and 3 black balls. A second bag contains 2 red and 4 black balls. One bag is selected at random. From the selected bag, one ball is drawn. Find the probability that the drawn ball is red.
To find the probability that the drawn ball is red, we need to consider two scenarios:
1. Selecting the first bag and drawing a red ball.
2. Selecting the second bag and drawing a red ball.
Let's denote the events:
A = Selecting the first bag
B = Selecting the second bag
R = Drawing a red ball
We need to find P(R), the probability that the drawn ball is red.
Using the law of total probability, we can express P(R) as:
P(R) = P(A) * P(R|A) + P(B) * P(R|B)
Now let's calculate each term:
P(A) = probability of selecting the first bag = 1/2 (since there are two bags and we are selecting one randomly)
P(R|A) = probability of drawing a red ball from the first bag = 4/7 (since there are 4 red balls out of 7 total in the first bag)
P(B) = probability of selecting the second bag = 1/2 (since there are two bags and we are selecting one randomly)
P(R|B) = probability of drawing a red ball from the second bag = 2/6 (since there are 2 red balls out of 6 total in the second bag)
Now plug in these values into the formula for P(R):
P(R) = (1/2) * (4/7) + (1/2) * (2/6)
= 2/7 + 1/6
Combining the terms with a common denominator:
P(R) = (2/7)*(6/6) + (1/6)*(7/7)
= 12/42 + 7/42
Adding the fractions:
P(R) = 19/42
Therefore, the probability that the drawn ball is red is 19/42.
To find the probability that the drawn ball is red, we need to consider two cases:
Case 1: Selecting the first bag and drawing a red ball.
Case 2: Selecting the second bag and drawing a red ball.
Let's calculate the probabilities for each case:
Case 1:
The probability of selecting the first bag is 1/2 since we have two bags to choose from.
Once the first bag is selected, the probability of drawing a red ball is 4/7, since there are 4 red balls out of a total of 7 balls in the bag.
Case 2:
Similarly, the probability of selecting the second bag is also 1/2.
Once the second bag is selected, the probability of drawing a red ball is 2/6, since there are 2 red balls out of a total of 6 balls in the bag.
To find the overall probability of drawing a red ball, we have to consider both cases and add them up:
Probability of drawing a red ball = (Probability of selecting the first bag)*(Probability of drawing a red ball from the first bag) + (Probability of selecting the second bag)*(Probability of drawing a red ball from the second bag)
= (1/2) * (4/7) + (1/2) * (2/6)
= 4/14 + 1/6
= 8/28 + 7/42
= (8 + 7)/42
= 15/42
Therefore, the probability of drawing a red ball is 15/42, which can be simplified to 5/14.
consider the result as 2 independent events
bag1, red
bag2, red
Prob(bag1,red) = (1/2)(4,7) = 2/7
prob(bag2,red) = (1/2)2/6) = 1/6
prob(your event) = 2/7 + 1/6 = 19/42