1.In a jar, suppose there are 3 blue marbles, 2 yellow marbles and one red marble.
c. Suppose that 3 marbles are drawn from the jar without replacement. What is the probability that the 3 marbles are all different colors?
d. Suppose that 4 marbles are drawn from the jar with replacement. What is the probability that exactly 2 of the four drawn are red?
e. Suppose that 3 marbles are drawn from the jar with replacement. How many yellow marbles would you expect to draw?
c) could be RBY RYB YBR YRB BRY OR BYR
prob of each one is (1/6)(3/5)(2/4) = 6/120 = 1/20
the others come out to the same result
so prob (all different) = 6/20 = 3/10
d) prob = C(4,2) (1/6)^2 (5/6)^2
= 6(1/36)(25/36) = 25/216
c. YOu can do it the following ways:
way Probability
B,Y,R Pr=3/6*2/5*1/4
R,B,Y =1/6*3/5*2/4
B,R,Y =3/6*1/5*2/4
R,Y,B
Y,R,B
Y,B,R
so the probability each time appears to be 3*2*1/6*5*4 so, with six ways, pr must be 6*3*2*1/(6*5*4) and you can reduce that.
Stop
c. To find the probability that the 3 marbles drawn are all different colors, we need to calculate the ratio of favorable outcomes to total outcomes.
Favorable outcomes: There are 3 choices for the first marble (blue, yellow, or red), 2 choices for the second marble (since we can't choose the same color as the first), and 1 choice for the third marble (since it can't be the same color as the first two).
Total outcomes: There are a total of 6 marbles in the jar, so we have 6 choices for the first marble, 5 choices for the second marble, and 4 choices for the third marble, assuming we never choose the same marble twice.
Therefore, the probability is (3 * 2 * 1) / (6 * 5 * 4) = 6 / 120 = 1 / 20, which simplifies to 0.05 or 5%.
d. To find the probability that exactly 2 of the four drawn marbles are red when drawing with replacement, we need to calculate the ratio of favorable outcomes to total outcomes.
Favorable outcomes: There are 6 marbles in the jar, with 1 red marble. Choosing exactly 2 out of the 4 marbles to be red can be done in (4 choose 2) = 6 different ways.
Total outcomes: Since we are drawing with replacement, each marble drawn is independent of the others, and there are 6 choices for each marble.
Therefore, the probability is 6 / (6 * 6 * 6 * 6) = 6 / 1296 = 1 / 216.
e. To calculate the expected number of yellow marbles when drawing 3 marbles with replacement, we first need to determine the probability of drawing a yellow marble each time.
Probability of drawing a yellow marble: There are 6 marbles in the jar, with 2 yellow marbles. So the probability of drawing a yellow marble on each draw is 2/6 = 1/3.
Since drawing each marble is independent, we can use the expected value formula: Expected value = Probability of event * Number of trials.
Expected number of yellow marbles = (Probability of drawing a yellow marble) * (Number of trials)
Expected number of yellow marbles = (1/3) * 3 = 1.
Therefore, you would expect to draw 1 yellow marble when drawing 3 marbles with replacement.