(a) An ordinary six-sided die numbered 1 to 6 is thrown. Write down the probability that the number shown on the die is a prime number.
A six-sided die numbered 1 to 6 and an eight sided die numbered 3 to 10 are thrown together. Giving each answer as a fraction in its lowest terms, find the probability that
(i) the sum of the numbers is 10,
(ii) the two numbers are not equal.
(b) A pupil has difficulty in waking up for school, and so, to wake himself, he sets 3 alarms to go off at the same time, as the noise from at least 2 alarms is necessary to wake him. Each alarm goes off independently.
The probability that each alarm goes off is 0.7, 0.8 and 0.9 respectively. Find the probability that
(i) all 3 go off,
(ii) the pupil is awakened.
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Answer:
(a) Prime numbers are 2, 3, 5
Hence, probability that it is a prime number = 3/6 = 1/2
(i) P(sum of numbers is 10)
= P(1,9) + P(2,8) + P(3,7) + P(4,6) + P(5,5) + P(6,4) => Summing all the possible combinations
= 1/6 * 1/8 * 6 times
= 1/8
(ii) P(two numbers are not equal)
= 1 - P(two numbers are equal)
= 1 - {P(3,3) + P(4,4) + P(5,5) + P(6,6)}
= 1 - {1/6 * 1/8 * 4 times}
= 1 - 1/12
= 11/12
(b)
(i) Probability that all 3 go off
= 7/10 * 8/10 * 9/10
= 63/125
(ii) P(Student is awakened)
= P(at least 2 alarms goes off)
= P(1,2, 3 don't go off) + P(2,3, 1 don't go off) + P(1,3, 2 don't go off) + P(all 3 go off) ==> P(which alarm goes off)
= 0.7 * 0.8 * 0.1 + 0.8 * 0.9 * 0.3 + 0.7 * 0.9 * 0.2 + 63/125
= 0.398 + 63/125
= 451/500
=
