(revised 4/5/96)
John C. Turner
1. In some gambling games, if you bet $1 and win, you receive $1.90 (for a net gain of $0.90). If you lose, you get nothing.
Suppose that winning and losing are equally likely. If I play this game 10 times, what is the chance that I have not lost money?
[Need 6 or more wins to make money. 1-P(<=5 wins)=1-0.62304=0.376953]
If I play the game 50 times? 100 times?
[50 plays requires 27 or more wins. 1-P(<=26 wins) = 1-0.62304 =0.3359 by BCDF.
By ZCDF, 1-0.66431 = 0.33568.
For 100 plays, need to win 53 or more times.
By ZCDF, 1-P(<=52 wins) = 1-0.69146 = 0.308537.]
2. I really need to win an additional $6 (over what I have in my pocket to start).
What is the probability of this happening if I play 10 times?
[0.0107]
What if I play 50 times? 100 times?
[0.101319, 0.135666]
3. The size of a litter of puppies has the following probability distn:
1 0.10 2 0.20 3 0.25 4 0.25 5 0.20
(Litters bigger than 5 can be ignored.) Suppose I have 30 pregnant bitches. How many puppies should I be prepared for?
At least (with a certain probability) how many? At most how many?
[The mean size of a litter is 3.25 and the variance is 1.5875.
The mean size of 30 litters is 97.5.
I am 90% sure the total will be less than 106.33 and 90% sure it will
be at least 88.666]
4. The time it takes a copier repairman to fix a copier has a normal distn with mean 2 hrs and std.dev. 0.2 hrs.
If there is only one hour left until quitting time, should I send him on another job?
[No. 1 hour is 5 s.d. less than the mean. It won't
happen.]
If I want to be 80% sure he can finish today, how much time should I allow?
[2.169 hrs]
He went on 10 jobs this week and his average time to finish was 2.1 hrs. Is this unusual? What if he averaged
2.1 hrs for 20 jobs? For how many jobs would there be a 90% chance that his average would be less than 2.1 hrs?
[10 jobs, P(avg<2.1)=0.94307
20 jobs, P(avg<2.1)=0.9873
For 6.55 jobs, P(avg<2.1)=0.90]
5. In surveying, we measure from point A to point B by measuring a sequence of "waypoints" and summing the results. The errors
would tend to accumulate. Suppose that each waypoint measurement has mean error of 0 and std.dev. of 0.6 ft.
If I took 10 waypoints to get from point A to point B, what is the probability that my total
error (in absolute value) is less than 1 ft? less than 2 ft?
I claim that 90% of the time I am within how much of the correct value?
Repeat for 20 waypoints.
6. We are going to load seabags on a cart. The cart can hold 650 lbs. The weight of each bag has a normal distribution with mean 50 lbs
and std.dev. of 10 lbs. What is the probability that the cart can hold 10 bags? 12 bags? 15 bags?
[10 bags with prob 0.999 12 0.9255 15 0.0049]
What if the std.dev. were 15 lbs instead of 10 lbs?
[10 bags with prob 0.9992 12 0.8320 15 0.0426]
For safety reasons, the number of bags I can load on the cart can only have a 1% chance of overloading it. How many bags can I load?
Do for std.dev. equal to both 10 and 15 lbs.
[std.dev=10 -> 11 bags is safe with prob 0.9987
std.dev=15 -> 10 bags is safe with prob 0.9778]
7. The time in the 100 yd dash has a normal distn with mean 12 sec and std.dev. 2 sec. How many can do the 100 yd dash in under 10 sec?
[0.158655]
If I run my students through in pairs, what is the chance that the faster student finishes in under 10 sec?
[0.29213]
If I run my entire class of 20 simultaneously, what is the chance that the fastest student finishes in under 10 sec?
[0.968]
What size class has a 50% chance that the fastest runner is under 10 secs?
[Class of 5. prob=0.57843]
8. The yield from a 10 acre field has a normal distn with mean 75 bushels and std.dev. 5 bushels.
How many fields produce more than 80 bushels?
[0.156855]
If my farm has 5 such 10 acre fields, what is the probability that my farm will produce more than 400 (=5*80) bushels?
[0.012674]
I need a silo to hold my produce. How big should my silo be so that I am 95% sure it will hold my crop?
[394 bushels prob=0.955]
10 farmers have formed a cooperative. (All their plots are as described above.) Instead of individual silos, they will build a
single grain elevator. How big should it be to be 95% sure of holding the entire harvest?
[3808 bushels with prob 0.949548]
9. Rework the gamma problems using the Central Limit
Thm. How close are your answers?
10. A car transport carries 8 cars. The weight of each car has a normal distn with mean 1500 lbs and std.dev. 100 lbs.
We designed the transport to be able to carry 14,000 lbs (=8*1500+safety margin).
What is the chance that the transport will be overloaded?
[VIrtually never]
What safety margin should we use to be 99% sure of being safe?
[12470 is safe with prob 0.9517]
11. Gallon milk bottles are filled to be accurate to within 1%.
(Take this to mean that 0.01 gal == 2 s.d.).
If I empty 20 bottles into a vat, how much will I have?
[certain to have 20 +/- 0.2
97.5% sure to have 20 +/- 0.1]
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