# [Probability] There are 50 cards of 5 different colors. It comprises of 10 Red cards, 10 blue cards, 10 orange cards, 10 green cards and 10 yellow cards. Each color will have the cards numbered between 1 to10. You pick 2 cards at random. What is the probability that they are not of same color and not of same number.

Data Science Interview QuestionsCategory: Data Science[Probability] There are 50 cards of 5 different colors. It comprises of 10 Red cards, 10 blue cards, 10 orange cards, 10 green cards and 10 yellow cards. Each color will have the cards numbered between 1 to10. You pick 2 cards at random. What is the probability that they are not of same color and not of same number.
MockInterview Staff asked 7 years ago

Lets say we pick the first card as a Red and 10. Now there are 49 cards remaining out of which 9 cards have the color red and 4 cards have the number 10 i.e. 13 cases that fail our condition of \”not of same color and not of same number\”. Hence probability of success = (49-13)/49 = 36/49

MockInterview Staff answered 7 years ago

So here’s the way I’m thinking about this one. We’re drawing two cards and we want to make sure they’re not the same number or same color. So the first card we draw can be anything (e.g. lets say, blue and number 5). It’s the second card that we have to worry about…

And for the second card, we’re calculating the P(not blue and not 5). Because these two events aren’t independent, P(not blue and not 5)=P(not blue)* P(not 5 | not blue). P(not blue) is 40/49. P(not 5 | not blue) is 36/40 because there are four 5’s left, and we’re only looking at the subset of cards that are not blue (40 cards). Therefore, the answer should be .734 (36/40 * 40/49)
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Zahar Hilkevich answered 6 years ago

There are C(50,2)=50*49/2!=25*49=1225 ways to pick 2 cards. If the 1st card is chosen (50 ways), there are 36 cards left that will make up the desired pair, so there are 50*36/2!=25*36 desired outcomes. The P=(25*36)/(25*49)=36/49

The Number of combinations for selecting two cards is (50 C 2) = 25*49— Which goes in Denominator of the Probability
Now to evaluate the number of favorable combinations, we need to select 2 colors from the given 5 colors i.e. (5 C 2), and from each of the 10 color set cards we need to select 1 card each i.e. (10 C 1) combinations, ass all the 3 events are independent the number of favorable combinations = (5 C 2)*(10 C 1)*(10 C 1) = 25*40
Therefore, The probability that we select two cards of different color = (25*40)/(25*49) = 40/49

Prob of getting two different color cards = 40/49 Prob of getting two different number cards = 45/49 Prob of getting different color and number = 40/49 * 45/49 =0.74