Diane Bruns is the mayor of a large city. Lately, she has become concerned about the possibility that large numbers of people who are drawing unemployment checks are secretly employed. Her assistants estimate that 40 percent of unemployment beneficiaries fall into this category, but Ms. Bruns is not convinced. She asks one of her aides to conduct a quiet investigation of 10 randomly selected unemployment beneficiaries. (a) If the mayor’s assistants are correct, what is the probability that more than eight of the individuals investigated have jobs? (Do not use the tables.) (b) If the mayor’s assistants are correct, what is the probability that one or three of the investigated individuals have jobs? (Do not use the tables.)
Answers
Given : 40 percent of unemployment beneficiaries who are drawing unemployment checks are secretly employed. conduct a quiet investigation of 10 randomly selected unemployment beneficiaries
To find : probability that more than eight of the individuals investigated have jobs
probability that one or three of the investigated individuals have jobs
Solution:
mayor’s assistants are correct,
=> probability of unemployment beneficiaries secretly employed p = 40/100 = 0.4
not secretly employed q = 1 -0.4 = 0.6
n = 10 ( number of samples)
P(x) = ⁿCₓpˣqⁿ⁻ˣ
probability that more than eight of the individuals investigated have jobs
= P(8) + P(9) + P(10)
= ¹⁰C₈(0.4)⁸(0.6)² + ¹⁰C₉(0.4)⁹(0.6)¹ +¹⁰C₁₀(0.4)¹⁰(0.6)⁰
= (0.4)⁸ ( 45 * (0.6)² + 10 * (0.4)* 0.6 + (0.4)²)
=(0.4)⁸ (16.2 + 2.4 + 0.16)
= (0.4)⁸ (18.76)
= 0.0123
probability that one or three of the investigated individuals have jobs
= P(1) + P(3)
= ¹⁰C₁(0.4)¹(0.6)⁹ + ¹⁰C₃(0.4)³(0.6)⁷
= (0.4)(0.6)⁷ ( 10 * (0.6)² + 120(0.4)²)
= (0.4)(0.6)⁷ ( 3.6 + 19.2)
= (0.4)(0.6)⁷ ( 22.8)
= 0.2553
0.0123 is the probability that more than eight of the individuals investigated have jobs
0.2553 is the probability that one or three of the investigated individuals have jobs
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