3. A bag contains 4 white, 3 blue, and 6 red marbles. A marble is drawn from the bag, replaced, and another marble is drawn. Find the probability that:
a) both marbles are red
b) both marbles are blue
c) the first marble is red and the second is blue
d) neither is red
To find the probability of each event, we need to consider the total number of possible outcomes and the number of favorable outcomes for each event.
a) Probability that both marbles are red:
Since there are 6 red marbles in the bag, the probability of drawing a red marble on the first draw is 6/13 (6 red marbles out of total 13 marbles).
Since the marble is replaced back into the bag, the probability of drawing a red marble again on the second draw is also 6/13.
To find the probability of both events happening, we multiply the probabilities:
P(both red) = P(red on first draw) × P(red on second draw) = (6/13) × (6/13) = 36/169
b) Probability that both marbles are blue:
Since there are 3 blue marbles in the bag, the probability of drawing a blue marble on the first draw is 3/13.
Since the marble is replaced back into the bag, the probability of drawing a blue marble again on the second draw is also 3/13.
To find the probability of both events happening, we multiply the probabilities:
P(both blue) = P(blue on first draw) × P(blue on second draw) = (3/13) × (3/13) = 9/169
c) Probability that the first marble is red and the second is blue:
The probability of drawing a red marble on the first draw is 6/13.
Since the marble is replaced back into the bag, the probability of drawing a blue marble on the second draw is 3/13.
To find the probability of both events happening, we multiply the probabilities:
P(red first and blue second) = P(red on first draw) × P(blue on second draw) = (6/13) × (3/13) = 18/169
d) Probability that neither is red:
The opposite event to the case where neither is red is that both marbles are red or at least one marble is red. We can find this probability and then subtract it from 1.
P(neither is red) = 1 - P(both red) = 1 - 36/169 = 133/169
To find the probability of events, we need to determine the number of favorable outcomes and the total number of possible outcomes. Let's calculate the probabilities for each event step by step:
a) Both marbles are red:
The probability of drawing a red marble on the first draw is 6/13, as there are 6 red marbles out of a total of 13 marbles. Since we are replacing the marble back after drawing, the probability of drawing another red marble on the second draw is also 6/13. To find the probability of both events happening, we multiply the probabilities together: (6/13) * (6/13) = 36/169.
b) Both marbles are blue:
The probability of drawing a blue marble on the first draw is 3/13, as there are 3 blue marbles out of a total of 13 marbles. Again, since we are replacing the marble back after drawing, the probability of drawing another blue marble on the second draw is also 3/13. To find the probability of both events happening, we multiply the probabilities: (3/13) * (3/13) = 9/169.
c) The first marble is red and the second is blue:
The probability of drawing a red marble on the first draw is 6/13. For the second draw, since we are placing the first marble back before the second draw, we still have 13 marbles in the bag, but now only 3 of them are blue. Therefore, the probability of drawing a blue marble on the second draw is 3/13. To find the probability of both events happening, we multiply the probabilities: (6/13) * (3/13) = 18/169.
d) Neither is red:
The probability of not drawing a red marble on the first draw is 1 - (6/13) = 7/13. Similarly, for the second draw, the probability of not drawing a red marble is 7/13 as well. To find the probability of both events happening, we multiply the probabilities: (7/13) * (7/13) = 49/169.
Therefore, the probabilities are:
a) Probability of both marbles being red = 36/169
b) Probability of both marbles being blue = 9/169
c) Probability of the first marble being red and the second being blue = 18/169
d) Probability of neither marble being red = 49/169