A bag contains 8 red marbles, 5 white marbles, and 6 blue marbles. You draw 4 marbles out at random, without replacement. Find the following probabilities and round to 4 decimal places.
a. The probability that all the marbles are red is
b. The probability that none of the marbles are red is
prob(all 4 red) = (8/19)(7/18)(6/17)(5/16) = 35/1935
or
prob(all red) = C(8,4)/C(19,4) = 70/3876 = 35/1938
Prob(none red) = C(11,4)/C(19,4) = 330/3876 = 55/646
or
prob(none red) = (11/19)(10/18)(9/17)(8/16) = 55/646
To find the probabilities, we need to calculate the ratio of favorable outcomes to the total number of possible outcomes.
a. The probability that all the marbles are red:
First marble: There are 8 red marbles and 19 total marbles, so the probability of drawing a red marble on the first draw is 8/19.
Second marble: After drawing one red marble, there are now 7 red marbles remaining and 18 total marbles, so the probability of drawing another red marble on the second draw is 7/18.
Third marble: After drawing two red marbles, there are now 6 red marbles remaining and 17 total marbles, so the probability of drawing another red marble on the third draw is 6/17.
Fourth marble: After drawing three red marbles, there are now 5 red marbles remaining and 16 total marbles, so the probability of drawing another red marble on the fourth draw is 5/16.
The probability of drawing all red marbles is calculated by multiplying the probabilities of each individual draw:
P(all red) = (8/19) * (7/18) * (6/17) * (5/16) ≈ 0.0317 (rounded to 4 decimal places)
Therefore, the probability that all the marbles are red is approximately 0.0317.
b. The probability that none of the marbles are red:
The total number of non-red marbles is 5 white marbles + 6 blue marbles = 11 non-red marbles.
First marble: There are 11 non-red marbles and 19 total marbles, so the probability of drawing a non-red marble on the first draw is 11/19.
Second marble: After drawing one non-red marble, there are now 10 non-red marbles remaining and 18 total marbles, so the probability of drawing another non-red marble on the second draw is 10/18.
Third marble: After drawing two non-red marbles, there are now 9 non-red marbles remaining and 17 total marbles, so the probability of drawing another non-red marble on the third draw is 9/17.
Fourth marble: After drawing three non-red marbles, there are now 8 non-red marbles remaining and 16 total marbles, so the probability of drawing another non-red marble on the fourth draw is 8/16.
The probability of drawing none of the marbles as red is calculated by multiplying the probabilities of each individual draw:
P(none red) = (11/19) * (10/18) * (9/17) * (8/16) ≈ 0.1069 (rounded to 4 decimal places)
Therefore, the probability that none of the marbles are red is approximately 0.1069.
To find probabilities, you need to use the formula:
P(event) = Number of favorable outcomes / Total number of possible outcomes
a. The probability that all the marbles drawn are red:
First, let's determine the total number of marbles in the bag: 8 red + 5 white + 6 blue = 19 marbles.
To find the probability of drawing all red marbles, you'll need to find the number of favorable outcomes and the total number of possible outcomes.
Number of favorable outcomes: Since you want to draw all red marbles, there are 8 red marbles in the bag. For the first draw, there are 8 red marbles out of 19 total marbles. After the first red marble is drawn and not replaced, there are now 7 red marbles out of 18 total marbles for the second draw. Similarly, for the third and fourth draws, the number of favorable outcomes also decreases. So the number of favorable outcomes is: 8/19 * 7/18 * 6/17 * 5/16.
Total number of possible outcomes: For each draw, you are picking from the total number of marbles, which decreases each time. So the total number of possible outcomes is: 19/19 * 18/18 * 17/17 * 16/16.
Now, we can plug in the values into the formula:
P(all red marbles) = (8/19) * (7/18) * (6/17) * (5/16)
P(all red marbles) ≈ 0.0301 (rounded to 4 decimal places)
b. The probability that none of the marbles drawn are red:
Similar to part a, you'll need to find the number of favorable outcomes and the total number of possible outcomes.
Number of favorable outcomes: In this case, you want to draw none of the red marbles. So the favorable outcomes will be the opposite of drawing all red marbles. This means you'll draw marbles from the white and blue ones in the bag. There are 5 white marbles and 6 blue marbles for the first draw, with decreasing numbers for the subsequent draws. So the number of favorable outcomes is: 5/19 * 4/18 * 3/17 * 2/16.
Total number of possible outcomes: Similar to part a, for each draw, you are picking from the total number of marbles, which decreases each time. So the total number of possible outcomes is: 19/19 * 18/18 * 17/17 * 16/16.
Now, use the formula again:
P(none of the marbles are red) = (5/19) * (4/18) * (3/17) * (2/16)
P(none of the marbles are red) ≈ 0.0144 (rounded to 4 decimal places)
Therefore:
a. The probability that all the marbles are red is approximately 0.0301.
b. The probability that none of the marbles are red is approximately 0.0144.