A probability question: choosing 3 from 25

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In summary: The probability of selecting 1 girl and 2 boys is (15 C 2) * (10 C 1)/ (25 C 3). This solution is found using the fundamental counting principle and represents the number of favorable outcomes divided by the total number of outcomes. The reason we multiply the number of ways to choose 1 girl and 2 boys is because we require both events to happen, and the product accounts for all possible combinations.
  • #1
first21st
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In a class, there are 15 boys and 10 girls. Three students are selected at random. The probability that 1 girl and 2 boys are selected, is:
A. 21/46
B. 25/117
C. 1/50
D. 3/25

Could you please solve this problem with proper explanation?

Thanks,

James
 
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  • #2
Re: A probability question

Here at MHB, we normally don't provide fully worked solutions, but rather we help people to work the problem on their own. This benefits people much more, which is our goal. (Nod)

Now, what you want to do here is to find the number of ways to choose 1 girl from 10 AND 2 boys from 15, then divide this by the number of ways to choose 3 children from 25. What do you find?
 
  • #3
Re: A probability question

Thanks for your reply. It's really a great way of learning. I highly appreciate your approach. Is the solution something like:

(15 C 2) * (10 C 1)/ (25 C 3)If it is correct, could you please explain why did we divide it with the number of ways of choosing 3 students from 25? Thanks,

James
 
  • #4
Re: A probability question

Excellent! That is correct! (Cool)

Now you just need to simplify, either by hand or with a calculator.

As probability is the ratio of the number favorable outcomes to the number of all outcomes, we are in this case dividing the number of ways to choose 1 girl from 10 AND 2 boys from 15 by the number of ways to choose 3 of the children from the total of 25. We are told that 3 children are selected at random, and we know there are 25 children by adding the number of boys to the number of girls. Thus, \(\displaystyle {25 \choose 3}\) is the total number of outcomes.
 
  • #5
Re: A probability question

Thank you very much man! But I am still wondering why did we MULTIPLY # the number of ways to choose 1 girl from 10 # AND # 2 boys from 15 #, instead of ADDING these two operands?
 
  • #6
Re: A probability question

We multiply because we are essentially applying the fundamental counting principle. When we require event 1 AND event 2 to happen, we multiply. When we require event 1 OR event 2 to happen, we add. Here we require both 2 boys AND 1 girl.

You see, for each way to obtain 2 boys, we have to account for all of the ways to obtain 1 girl. Or conversely, for each way to obtain 1 girl, we have to account for all of the ways to obtain 2 boys. The product of these two gives us all of the ways to get 2 boys and 1 girl.
 
  • #7
Re: A probability question

ok. Got it! Thanks a lot.James
 

FAQ: A probability question: choosing 3 from 25

1. What is the probability of choosing 3 items from a set of 25?

The probability can be calculated as (n choose k) / (N choose K), where n is the number of desired items, k is the number of items to choose, N is the total number of items in the set, and K is the total number of items to choose from. In this case, it would be (3 choose 25) / (3 choose 25) = 0.00002, or 0.002%.

2. How do you calculate the number of distinct combinations when choosing 3 items from a set of 25?

The number of distinct combinations can be calculated using the formula n! / (k!(n-k)!), where n is the total number of items and k is the number of items to choose. In this case, it would be 25! / (3!(25-3)!) = 2300 distinct combinations.

3. Is there a difference between choosing 3 items from 25 in a specific order or without a specific order?

Yes, there is a difference. Choosing 3 items from 25 in a specific order means that the order in which the items are chosen matters, while choosing without a specific order means that the items can be chosen in any order. The number of distinct combinations will be different in each case.

4. What is the probability of choosing 3 items from a set of 25 without replacement?

The probability without replacement can be calculated as (n choose k) / (N choose K), where n is the number of desired items, k is the number of items to choose, N is the total number of items in the set, and K is the total number of items to choose from. However, in this case, the value of N will decrease by 1 for each item chosen. So, for the first item, it would be (3 choose 25) / (3 choose 25) = 0.00002. For the second item, it would be (2 choose 24) / (2 choose 24) = 0.00008. And for the third item, it would be (1 choose 23) / (1 choose 23) = 0.001. Therefore, the overall probability would be 0.00002 x 0.00008 x 0.001 = 0.000000000016, or 0.0000016%.

5. How does the probability change if we increase the number of items to choose from?

If we increase the number of items to choose from, the probability of choosing 3 items from the set will decrease. This is because the total number of possible combinations will increase, making it less likely for our desired combination of 3 items to be chosen. However, the exact change in probability will depend on the specific number of items added to the set.

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