A pair of fair dice is rolled. Find the probability that the sum of the two numbers facing up is less than 10.
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Let the combinations of numbers that total up sums less than 10 are : (x, y)
x is the number facing up on the first dice.
y is the number facing up on the second dice.
Let p = probability that x + y < 10 = P(x+y <10)
= 1 - P (x+y>= 10)
= 1 - P [ (x = 4 AND y = 6 ) OR ( x = 5 AND y = 5 ) OR (x=6 AND y = 4) ]
= 1 - [ P(4,6) + P(5,5) + P(6,4) ]
= 1 - [ 1/6 * 1/6 + 1/6*1/6 + 1/6 * 1/6 ]
= 1 - 3/36
= 11/12
the probabilities are multiplied, above, as they (results of two dice rolling) are independent events.
x is the number facing up on the first dice.
y is the number facing up on the second dice.
Let p = probability that x + y < 10 = P(x+y <10)
= 1 - P (x+y>= 10)
= 1 - P [ (x = 4 AND y = 6 ) OR ( x = 5 AND y = 5 ) OR (x=6 AND y = 4) ]
= 1 - [ P(4,6) + P(5,5) + P(6,4) ]
= 1 - [ 1/6 * 1/6 + 1/6*1/6 + 1/6 * 1/6 ]
= 1 - 3/36
= 11/12
the probabilities are multiplied, above, as they (results of two dice rolling) are independent events.
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