Laurel appears at Comedy Cafe at a random time between 1:00 and 2:00. Hardy turns up at a random time between 2:00 and 2:30. The probability that Laurel waits for no more than 40 minutes for Hardy can be expressed in the form m/n, where m and n are relatively prime positive integers. Find m + n.
Events:
S=Laurel waits 40 minutes or less for hardy.
~S=Laurel waits more than 40 minutes.
Assumptions: both arrivals are random and create a uniform probability distribution.
Between 1:00 and 1:20, Laurel will be sure to wait 40 minutes or more, since Hardy does not arrive before 2:00. So P(S)=0
Between 1:50 and 2:00, Laurel will always wait 40 minutes or less for Hardy, so P(S)=1.
Between 1:20 and 1:50, the probability of success increases from 0 to 1, i.e. this is the period where the 40 minutes covers more and more of the half-hour that Hardy arrives.
If we sum the three areas and divide by the duration (60 minutes), we have the probability that Laurel would have waited 40 minutes or less for Hardy.
A=0*20+(1/2)(0+1)*30+10*1
=0+15+10
=25
Over the one hour Laurel could arrive,
P(S)=25/60=5/12
=>
m=5, n=12, m+n=5+12=17
1 .....
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0........
A sketch of the probability distribution.
Sorry the sketch did not work out! :(
Well, it seems like Laurel and Hardy have quite the comedic timing! Let me think about this for a moment...
We know that Laurel arrives between 1:00 and 2:00, and Hardy arrives between 2:00 and 2:30. So, let's break down the possibilities and calculate the probability:
1) Laurel arrives at 1:00:
- In this case, Hardy has a 40-minute window between 2:00 and 2:40 to arrive. The probability of this happening is (40)/(30) = 4/3.
2) Laurel arrives between 1:01 and 1:20:
- In this scenario, Hardy has exactly 40 minutes before he arrives. So, the probability is 1.
3) Laurel arrives between 1:21 and 1:30:
- Here, Hardy only has a window of 10 minutes before he arrives. Probability = (10)/(30) = 1/3.
4) Laurel arrives between 1:31 and 1:40:
- Now, Hardy has no time left to arrive. So, the probability is zero.
Adding up these probabilities gives us:
(4/3) + 1 + (1/3) + 0 = 8/3.
Therefore, m = 8 and n = 3. And finally, m + n = 8 + 3 = 11.
So, the sum m + n is 11.
To find the probability that Laurel waits for no more than 40 minutes for Hardy, we need to calculate the favorable outcomes divided by the total number of possible outcomes.
First, let's determine the total number of possible outcomes:
- The range of possible arrival times for Laurel is between 1:00 and 2:00, which is a duration of 60 minutes.
- The range of possible arrival times for Hardy is between 2:00 and 2:30, which is a duration of 30 minutes.
Therefore, the total number of possible outcomes is 60 * 30 = 1800.
Now, let's determine the favorable outcomes:
- For Laurel to wait no more than 40 minutes for Hardy, Laurel must arrive between 1:00 and 1:20.
- Thus, the favorable outcomes are (20 minutes) * (30 minutes) = 600.
Finally, let's calculate the probability:
Probability = favorable outcomes / total outcomes = 600 / 1800 = 1/3.
Therefore, the probability that Laurel waits for no more than 40 minutes for Hardy is 1/3.
To find m + n, we have m = 1 and n = 3. So, m + n = 1 + 3 = 4.
To find the probability that Laurel waits for no more than 40 minutes for Hardy, we should analyze the possible time intervals when Laurel and Hardy could arrive.
1. Laurel's arrival time: Between 1:00 and 2:00 (60 minutes)
2. Hardy's arrival time: Between 2:00 and 2:30 (30 minutes)
To solve this problem, we will consider the options for both Laurel and Hardy's arrival times and determine the intervals where Laurel would wait no more than 40 minutes for Hardy.
Case 1: Laurel arrives before 1:20
If Laurel arrives before 1:20, she has to wait for Hardy to arrive between 2:00 and 2:20 (40 minutes). The probability of this happening is (20/60) = 1/3.
Case 2: Laurel arrives between 1:20 and 1:40
If Laurel arrives between 1:20 and 1:40, she has to wait for Hardy to arrive between 2:00 and 2:30 (30 minutes). The probability of this happening is (30/60) = 1/2.
Case 3: Laurel arrives between 1:40 and 2:00
If Laurel arrives between 1:40 and 2:00, she has to wait for Hardy to arrive between 2:00 and 2:30 (30 minutes). The probability of this happening is (30/60) = 1/2.
Now, we need to calculate the overall probability by considering all three cases.
Probability = (1/3) * (1/2) + (1/2) * (1/2) + (1/2) * (1/2)
Probability = 1/6 + 1/4 + 1/4
Probability = 5/12
Therefore, the probability that Laurel waits for no more than 40 minutes for Hardy is 5/12.
To find m + n, we add the numerator and denominator of the fraction together:
m + n = 5 + 12 = 17
Hence, the value of m + n is 17.