Solving Probability Problem: 11/20 x 9/19 = 266/380

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In a class of 20 pupils with varying hair colors, the probability of selecting two pupils with different hair colors is calculated to be 266/380. The initial incorrect calculation of 99/380 arose from not ensuring that the second pupil chosen had a different hair color than the first. To find the correct probability, it is suggested to calculate the complementary event, which is the probability that both pupils have the same hair color. This involves calculating the probabilities for each hair color and summing them. The discussion emphasizes the importance of considering the total number of remaining pupils and their hair colors when calculating probabilities.
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Can anyone help me with this?

In a class of 20 pupils, 11 have dark hair, 7 have blond hair and 2 have red hair. 2 Pupils are chosen at random to collect the homework. What is the probability that each have a different colour hair?

The correct answer is 266/380

However I thought it would be 99/380. I arrived at that answer in the following way:

11/20 x 9/19 = 99/380

Where did I go wrong?
 
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In your setup you don't ensure the second person has hair of a different color than the first.
 
Hi. Thanks for your reply but I'm afraid I don't understand. Would you mind laying it out for me as it should be done?
Thanks again for your help.
 
Well, I assume you are aware on how to calculate the probability of the complementary event?

In short, try to calculate the probability that the 2 chosen pupils have the same hair colour, which should be easier to deal with.
 
The probability that the first student has black hair is 11/20 and the probability that the second student does NOT is 9/19 (because there are now 19 students left and the first had black hair so all blond and redheaded students are still left.

The probability that the first student has blonde hair is 7/20 and the probability that the second student does NOT is 13/19.

The probability that the first student has red hair is 2/20 and the probability that the second student does NOT is 18/19.

The probability that "A and B" happen is the product of probabilities of A and B separately and the probability that one of three different things happen is their sum.
 

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