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Math Help - Conditional probability

  1. #1
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    Conditional probability

    I'm a bit confused about this problem and was hoping someone sould give me a push in the right direction

    Sarah and Thomas each roll a die. Whoever rolls the higher number wins; if they both roll the same number, neither wins
    a.What is the probability that Thomas wins?
    b.If Sarah rolls a 3, what is the probability that she wins?
    c.If Sarah rolls a 3, what is the probability that Thomas wins?
    d.If Sarah wins, what is the probability that Thomas rolled a 3
    e.If Sarah wins, what is the probability that Sarah rolled a 3?

    Thanks
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  2. #2
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    Keep in mind that for a situation which calls for the rolling of two six-sided dice, there are 36 possible outcomes.

    a) There are 6 situations in which both players roll the same number and neither wins. That leaves 30 situations in which each player rolls a different number. These 30 situations come in pairs: Given x > y, Sarah can roll x and Thomas can roll y, or Sarah can roll y and Thomas can roll x. Because of this symmetry, there are 15 situations (half of 30) in which Thomas wins. The probability of Thomas winning is 15/36, or 5/12.

    b) Given that Sarah rolls a 3, there are only 6 possible outcomes. In two of them, Thomas rolls lower than a 3 and Sarah wins. In the other four, Thomas rolls at least a 3 and Sarah does not win. The probability of Sarah winning is 2/6, or 1/3.

    c) In the same situation as b), Thomas can win if he rolls higher than a 3, which can happen if he rolls a 4, 5, or 6. So there are 3 outcomes in which Thomas wins. The probability of Thomas winning is 3/6, or 1/2.

    d) Given that Sarah wins, there are 15 possible outcomes. If Thomas rolls a 3 and Sarah wins, that means she must have rolled a 4, 5, or 6. So she can win three different ways when Thomas rolls a 3. The probability of Sarah winning and Thomas rolling a 3 in this situation is 3/15 or 1/5.

    e) Again, there are 15 possible outcomes. However, she can win with a 3 only when Thomas rolls a 1 or a 2, which is two ways. The probability of Sarah winning and rolling a 3 in this situation is 2/15.
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  3. #3
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    Quote Originally Posted by icemanfan View Post
    Keep in mind that for a situation which calls for the rolling of two six-sided dice, there are 36 possible outcomes.

    a) There are 6 situations in which both players roll the same number and neither wins. That leaves 30 situations in which each player rolls a different number. These 30 situations come in pairs: Given x > y, Sarah can roll x and Thomas can roll y, or Sarah can roll y and Thomas can roll x. Because of this symmetry, there are 15 situations (half of 30) in which Thomas wins. The probability of Thomas winning is 15/36, or 5/12.

    b) Given that Sarah rolls a 3, there are only 6 possible outcomes. In two of them, Thomas rolls lower than a 3 and Sarah wins. In the other four, Thomas rolls at least a 3 and Sarah does not win. The probability of Sarah winning is 2/6, or 1/3.

    c) In the same situation as b), Thomas can win if he rolls higher than a 3, which can happen if he rolls a 4, 5, or 6. So there are 3 outcomes in which Thomas wins. The probability of Thomas winning is 3/6, or 1/2.

    d) Given that Sarah wins, there are 15 possible outcomes. If Thomas rolls a 3 and Sarah wins, that means she must have rolled a 4, 5, or 6. So she can win three different ways when Thomas rolls a 3. The probability of Sarah winning and Thomas rolling a 3 in this situation is 3/15 or 1/5.

    e) Again, there are 15 possible outcomes. However, she can win with a 3 only when Thomas rolls a 1 or a 2, which is two ways. The probability of Sarah winning and rolling a 3 in this situation is 2/15.
    Thank you very much
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