A bag contains 5 red marbles, 4 yellow marbles, 3 green marbles, 2 orange marbles and 2 purple marbles. If three marbles are removed, what is the probability that at least one of them is red?
To find the probability that at least one of the three marbles drawn is red, we need to consider the total number of possible outcomes and the number of favorable outcomes.
Step 1: Find the total number of possible outcomes.
Since we are drawing three marbles from the bag, the total number of possible outcomes can be calculated using the concept of combinations. The formula for combinations is:
nCr = n! / (r!(n-r)!)
where n is the total number of marbles and r is the number of marbles being drawn.
In this case, we have a total of (5 + 4 + 3 + 2 + 2) = 16 marbles in the bag.
So, the total number of possible outcomes is:
16C3 = 16! / (3!(16-3)!) = 560
Step 2: Find the number of favorable outcomes.
To find the number of favorable outcomes, we need to consider the cases where at least one of the three marbles drawn is red. We can subtract the cases where none of the marbles drawn is red from the total possible outcomes.
Case 1: None of the three marbles drawn is red.
In this case, we need to choose all three marbles from the non-red marbles (all other colors except red).
Number of non-red marbles = (4 + 3 + 2 + 2) = 11
Number of outcomes for this case = 11C3 = 11! / (3!(11-3)!) = 165
Case 2: At least one of the three marbles drawn is red.
Number of favorable outcomes = Total possible outcomes - Outcomes in case 1
Number of favorable outcomes = 560 - 165 = 395
Step 3: Calculate the probability.
Finally, we can calculate the probability of drawing at least one red marble by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 395 / 560 = 0.7054 (rounded to four decimal places) or 70.54% (rounded to two decimal places)
Therefore, the probability that at least one of the three marbles drawn is red is approximately 0.7054 or 70.54%.
To find the probability that at least one of the three marbles removed is red, we need to calculate the complementary probability of no red marbles being selected.
Step 1: Calculate the total number of marbles in the bag
Total number of marbles = 5 (red) + 4 (yellow) + 3 (green) + 2 (orange) + 2 (purple) = 16 marbles
Step 2: Calculate the number of ways to select 3 marbles out of the 16 marbles
We can use combinations to calculate this. The formula for combinations is:
nCr = n! / (r! * (n-r)!)
where n is the number of items and r is the number of items to be chosen. In this case, n = 16 and r = 3.
16C3 = 16! / (3! * (16-3)!) = 16! / (3! * 13!) = (16 * 15 * 14) / (3 * 2 * 1) = 560
So, there are 560 ways to select 3 marbles from the bag.
Step 3: Calculate the number of ways to select 3 marbles with no red marbles
We need to choose all 3 marbles from the non-red marbles.
Number of non-red marbles = 4 (yellow) + 3 (green) + 2 (orange) + 2 (purple) = 11 marbles
Number of ways to choose 3 non-red marbles = 11C3 = 11! / (3! * (11-3)!) = (11 * 10 * 9) / (3 * 2 * 1) = 165
Step 4: Calculate the probability of no red marbles being selected
Probability of no red marbles = Number of ways to choose 3 non-red marbles / Number of ways to select 3 marbles
= 165 / 560
Step 5: Calculate the probability that at least one of the three marbles is red
Probability = 1 - Probability of no red marbles
= 1 - (165 / 560)
= 1 - 0.2946
= 0.7054
Therefore, the probability that at least one of the three marbles removed is red is approximately 0.7054 or 70.54%.
Since there are 5 red marbles and 11 others, then the chance is
11/16 * 10/15 * 9/14 = 33/112 that the first three draws are not red.
So, that leaves 79/112 chance that at least one was red.