Problem 10 · Conditional counting by cases · 5 points
Made by AI and not reviewed by a real person.
Four students are chosen at random without replacement. If their study hours fall in exactly three bars of the chart, what is the probability that the bar represented twice is 4 to under 6 hours or 6 hours or more?
- 303/724
- 23/59
- 8/19
- 1/2
- 7/16
Show a hint
- The repeated bar is not equally likely to be any of the four bars: its size also affects which choices remain for the other two students.
- For each possible repeated bar, count pairs from that bar, then choose the other two students from two different bars.
Show the answer and solution
Answer: A — 303/724
Count each four-student group once, according to its repeated bar. The numbers of valid groups when that bar is under 2 hours, 2 to under 4 hours, 4 to under 6 hours, and 6 hours or more are, respectively, 136,500; 337,125; 270,000; and 70,875. Thus the conditional probability is (270,000 + 70,875)/(136,500 + 337,125 + 270,000 + 70,875) = 303/724. The tempting 1/2 treats all four bars as equally likely to supply the repeated pair, but groups with different repeated bars occur in different numbers.