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In a box carrying one dozen of oranges, one-fourthhave become bad. If 3 oranges are taken out from the box at random, what is the probability that at least one orange out of the three oranges picked up is good?
- a)52/55
- b)219/220
- c)44/45
- d)54/65
- e)None of these
Correct answer is option 'B'. Can you explain this answer?
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To solve this problem, we need to find the probability that at least one orange out of the three picked is good. Let's break down the problem step by step.
Step 1: Calculate the total number of possible outcomes.
There are 12 oranges in the box, and we want to pick 3 of them. The total number of possible outcomes is given by the combination formula:
nCr = n! / (r!(n-r)!)
In this case, n = 12 (total number of oranges) and r = 3 (number of oranges picked). Plugging in the values, we get:
12C3 = 12! / (3!(12-3)!) = 12! / (3!9!) = (12*11*10) / (3*2*1) = 220
So, there are 220 possible outcomes.
Step 2: Calculate the number of favorable outcomes (picking at least one good orange).
To calculate the number of favorable outcomes, we need to consider two cases:
1. Picking one good orange and two bad oranges.
2. Picking two good oranges and one bad orange.
Case 1: Picking one good orange and two bad oranges.
There are 9 bad oranges in the box, and we want to pick 2 of them. The number of ways to do this is given by the combination formula:
9C2 = 9! / (2!(9-2)!) = 9! / (2!7!) = (9*8) / (2*1) = 36
Now, there is one good orange left in the box, and we want to pick 1 of them. The number of ways to do this is simply 1.
Therefore, the number of favorable outcomes for this case is 36 * 1 = 36.
Case 2: Picking two good oranges and one bad orange.
There are 3 good oranges in the box, and we want to pick 2 of them. The number of ways to do this is given by the combination formula:
3C2 = 3! / (2!(3-2)!) = 3! / (2!1!) = (3*2) / (2*1) = 3
Now, there are 8 bad oranges left in the box, and we want to pick 1 of them. The number of ways to do this is simply 8.
Therefore, the number of favorable outcomes for this case is 3 * 8 = 24.
Step 3: Calculate the probability.
The probability is given by the number of favorable outcomes divided by the total number of possible outcomes.
Probability = (number of favorable outcomes) / (total number of possible outcomes)
Probability = (36 + 24) / 220
Probability = 60 / 220
Probability = 3 / 11
Therefore, the probability that at least one orange out of the three picked is good is 3/11.
The correct answer is option B) 219/220
Step 1: Calculate the total number of possible outcomes.
There are 12 oranges in the box, and we want to pick 3 of them. The total number of possible outcomes is given by the combination formula:
nCr = n! / (r!(n-r)!)
In this case, n = 12 (total number of oranges) and r = 3 (number of oranges picked). Plugging in the values, we get:
12C3 = 12! / (3!(12-3)!) = 12! / (3!9!) = (12*11*10) / (3*2*1) = 220
So, there are 220 possible outcomes.
Step 2: Calculate the number of favorable outcomes (picking at least one good orange).
To calculate the number of favorable outcomes, we need to consider two cases:
1. Picking one good orange and two bad oranges.
2. Picking two good oranges and one bad orange.
Case 1: Picking one good orange and two bad oranges.
There are 9 bad oranges in the box, and we want to pick 2 of them. The number of ways to do this is given by the combination formula:
9C2 = 9! / (2!(9-2)!) = 9! / (2!7!) = (9*8) / (2*1) = 36
Now, there is one good orange left in the box, and we want to pick 1 of them. The number of ways to do this is simply 1.
Therefore, the number of favorable outcomes for this case is 36 * 1 = 36.
Case 2: Picking two good oranges and one bad orange.
There are 3 good oranges in the box, and we want to pick 2 of them. The number of ways to do this is given by the combination formula:
3C2 = 3! / (2!(3-2)!) = 3! / (2!1!) = (3*2) / (2*1) = 3
Now, there are 8 bad oranges left in the box, and we want to pick 1 of them. The number of ways to do this is simply 8.
Therefore, the number of favorable outcomes for this case is 3 * 8 = 24.
Step 3: Calculate the probability.
The probability is given by the number of favorable outcomes divided by the total number of possible outcomes.
Probability = (number of favorable outcomes) / (total number of possible outcomes)
Probability = (36 + 24) / 220
Probability = 60 / 220
Probability = 3 / 11
Therefore, the probability that at least one orange out of the three picked is good is 3/11.
The correct answer is option B) 219/220
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