if you have 8 marbles in the bag (5 blue and 3 green) and you draw 2 with replacement, what is the theoretical probability that both marbles wil be green
how does the probability of getting two greens change if you are not replacing??
if you have 8 marbles in the bag (5 blue and 3 green) and you draw 2 without replacement, what is the theoretical probability that both marbles will be green
if you have 8 marbles in the bag (5 blue and 3 green) and you draw 2 with replacement, what is the theoretical probability that both marbles will be green
idk
11% without
39% with
with replacement:
prob(2 green) = (3/8)(3/8) = 9/64
without replacement
prob(2 green) = (3/8)(2/7) = 6/56 = 3/28
To determine the theoretical probability of drawing marbles, we need to know the total number of outcomes and the favorable outcomes.
1. Drawing with Replacement:
In this case, the total number of outcomes is calculated by multiplying the number of marbles in the bag by the number of draws. Since there are 8 marbles in the bag and you draw 2 marbles with replacement, the total number of outcomes is 8 * 8 = 64.
To find the favorable outcomes, we need to consider that there are 3 green marbles in the bag. Since we are drawing with replacement, after drawing the first green marble, the total number of green marbles remains the same. Therefore, the favorable outcomes for drawing two green marbles are 3 * 3 = 9.
The theoretical probability of drawing two green marbles with replacement is calculated by dividing the favorable outcomes by the total number of outcomes: 9 / 64 = 0.1406 or 14.06%.
2. Drawing without Replacement:
In this case, the total number of outcomes is still calculated by multiplying the number of marbles in the bag by the number of draws. So, the total number of outcomes is 8 * 7 = 56 (since you don't replace the marbles after each draw).
The favorable outcomes for drawing two green marbles without replacement are calculated by multiplying the number of green marbles by the number of green marbles remaining in the bag after the first draw. So, the favorable outcomes are 3 * 2 = 6.
The theoretical probability of drawing two green marbles without replacement is calculated by dividing the favorable outcomes by the total number of outcomes: 6 / 56 = 0.1071 or 10.71%.
In summary:
- Drawing with replacement: Theoretical probability = 0.1406 or 14.06%
- Drawing without replacement: Theoretical probability = 0.1071 or 10.71%
In both cases, we see that the probability of drawing two green marbles is higher when drawing with replacement compared to without replacement. This is because when we draw with replacement, the composition of the bag remains the same after each draw, giving us a higher chance of drawing green marbles.