Questions and solutions on Probability
John looks at the clock at some arbitrary time between 4pm and 6 pm and notices that the angle between the two hands is an acute angle. He looks at the clock exactly 15 minutes later, what is the probability that during this observation also he sees that the angle between the hour and minute hands to be an acute-angle?
Correct Answer : (C)
At 4:00 pm the angle between the hands is 1200. The minutes hand is 1200 behind the hour hand. Now, let us first see when the angle will become an acute angle. The angle between he hands will be acute when the minutes hand gains more than 300. Or, after 30/330 * 60 minutes.
Somewhere between 4:05 and 4: 10.From now, the angle will be acute till the minute hand reaches a point where it is 90 degree ahead of the hour hand. Or, till the time the minute and gains 1800 (Going from 900 degree behind to 900 ahead of hour hand), the angle will be acute. Or, for a spell of 180/330 * 60 minutes, the angle will be acute. Or for a spell of 360/11 minutes, the angle will be acute. For a spell of 32 8/11 minutes the angle will be acute.
So, if John had seen the clock at a time from 4 pm to 5pm, he would have seen it during this spell. Now, if his second viewing had also happened within this spell, he would have seen an acute angle second time.
Or, t, t + 15 should have both been within this 32 8/11 range for him to have seen two acute angle observations. Or, t should have happened at least 15 minutes before the angle become obtuse again. If we have the time range for it being acute as (x, x + 32 8/11), The question can be restated as if t belongs to this range, what it is the probability that t + 15 also belongs to this range.
For t + 15 to be within the range, t + 15 < x + 32 8/11, or t < 32 8/11 – 15 = 17 8/11
Probability should be 195/11 divided by 360/11 = 195/360 = 39/72 = 13/24
In the 5pm to 6pm range we get an identical probability. So the overall probability is 13/24
Answer Choice (C)
Level of Difficulty 2