1. When 2 dice at thrown, what is the probability of the sum being 9? 2. Two cards are picked at random from a standard deck of cards. What is the probability of picking an ace and a queen?
When 2 dice are thrown, the probability of the sum being 9 is 1/9. When 2 cards are picked at random from a standard deck of cards, the probability of picking an ace and a queen is 8/663.
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Problem 1.
When a die is thrown the outcome can be any of the numbers from 1 to 6. If two dice are thrown the set of outcomes that ensure the sum is 9 is {(3, 6), (6,3), (4, 5), (5, 4)}. The total number of possible outcomes is 6*6 = 36
This gives the required probability as 4/36 = 1/9
The probability of getting 9 as the sum when 2 dice are thrown is 1/9.
Problem 2.
A standard deck of cards has 52 cards with 4 aces and 4 queens. The total number of outcomes when two cards are randomly picked is 52*51 = 2652. There are 4 ways in which the first card picked can be an ace. And there are 4 ways in which the second card can be a queen. It is important to notice that the order of picking an ace and a queen is not important here. As a result, we also have to consider the possibility of picking a queen followed by an ace. This gives the total number of ways in which an ace and a queen can be picked as 4*4 + 4*4 = 32. The required probability is 32/2652 = 8/663
When 2 cards are randomly picked from a standard deck of cards the probability of picking a queen and an ace is 8/663.
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1. In order to get a sum of 9 with two dice, you would have to roll the pairs 4 & 5, 5 & 4, 3 & 6, or 6 & 3. Since you have two dice there are 36 different combinations (6 faces*6 faces) you could roll, but only 4 of them would give you a sum of 9. That makes the probability 4/36 or 1/9.
2. There are 52 cards in a deck. If you draw two at random you'd have 52 to choose from for your first draw and only 51 for your second. That gives you 2,652 combinations of cards total. Out of the 52 cards in the deck, 4 are aces and 4 are queens. This means you can choose an ace and then a queen (4*4) or a queen and then an ace (4*4). You'll have 32 combinations of an ace and a queen total, giving you a probability of 32/2,652 or 8/663.
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