The Probability of drawing two black balls in succession from a bag containing 4 red and 3 black balls when the ball first drawn is

a) replaced
b) not replaced

a) if replaced, prob = (3/7)(3/7) = 9/49

b) if not replaced prob = (3/7)(2/6) = 1/7

a) When the ball is replaced after the first draw:

The probability of drawing a black ball on the first draw is 3/7, because there are 3 black balls out of a total of 7 balls in the bag.
Since the ball is replaced, the probability of drawing a black ball on the second draw is also 3/7, as the number of black balls and the total number of balls in the bag remain the same.
To find the probability of drawing two black balls in succession, we multiply the probabilities of both events: (3/7) * (3/7) = 9/49.

Therefore, the probability of drawing two black balls in succession, with replacement, is 9/49.

b) When the ball is not replaced after the first draw:
The probability of drawing a black ball on the first draw is still 3/7.
However, after removing one black ball from the bag, there are now only 2 black balls left out of a total of 6 balls (since one ball has been removed).
Therefore, the probability of drawing a black ball on the second draw, without replacement, is 2/6 = 1/3.
To find the probability of drawing two black balls in succession, we multiply the probabilities of both events: (3/7) * (1/3) = 3/21.

Therefore, the probability of drawing two black balls in succession, without replacement, is 3/21.

To find the probability of drawing two black balls in succession, we need to consider whether the ball is replaced or not after each draw.

a) When the ball is replaced after each draw:
In this case, the number of balls in the bag remains the same after each draw. There are a total of 4 red and 3 black balls, so the probability of drawing a black ball on the first draw is 3/7. Since the ball is replaced, the probability of drawing another black ball on the second draw is also 3/7. To find the probability of two successive events happening, we multiply the individual probabilities together:

Probability (drawing two black balls with replacement) = (3/7) * (3/7) = 9/49

b) When the ball is not replaced after each draw:
In this case, the number of balls in the bag changes after each draw. After drawing the first black ball, there will be one less black ball and one less ball overall in the bag. The probability of drawing a black ball on the first draw is 3/7. After that, there will be a total of 6 balls left, with 3 of them being black. Therefore, the probability of drawing a black ball on the second draw is 3/6. Again, we multiply the individual probabilities to get the probability of two successive events:

Probability (drawing two black balls without replacement) = (3/7) * (3/6) = 9/42 = 3/14

Therefore, the probability of drawing two black balls in succession from a bag containing 4 red and 3 black balls is:
a) With replacement: 9/49
b) Without replacement: 3/14