When my mom sufferend a catastrophic stroke, I returned to my home town and found a pool in which to swim. I swam mostly alone in a pool that was 50 meters by 25 meters. Sometimes the lane lines were set to allow swimmers to swim "short course" (25 meters) or "long course" (50 meters).
On one of the short course days, I was alone in the pool. There were 16 empty lanes. I chose lane 8, one of the middle lanes. I swam down-and-back (50 meters) in about 45 seconds. This became a deliciously long swim as I moved back and forth alone in this pool with my own personal lifeguard. Then, suddenly, I was jerked out of my fraction-calculating delirium (refer to previous post about Fractions in the Fast Lane) when waves overtook me. A man dropped himself and his large belly covered in fur into the lane next to mine at precisely the moment I was at his end of the pool.
Ah, I thought. A probability problem. What is the probability that he would (a) choose the lane next to mine and at the same time (b) choose to enter the water during the roughly 7 seconds that I am vulnerable to the tsunami he created at the near end of the pool. And is this probability small enough that I should think this individual inconsiderate?
On one of the long course days, I was again alone in the pool, now well spoiled and feeling like the Queen of Sheba in her own blue-glass lake (with lane-lines and a black line at the bottom.) There were 9 lanes. I chose the middle lane. I would swim 50 meters to the far end and 50 meters back, giving me new distances and fractions to consider. I swam the same rate as in the short course setting in the pool. Again, as I swam a deliciously long swim with new numbers flowing along side of me, another man, a more narrow man, entered the pool. Now then, many of us in this world like to be individuals; we value our ability to make choices and to be unique. This man decided to swim a uniquely different way in this pool. He wanted to swim 25 meters, not 50 meters as the pool was set. So he did, weaving his way across the pool, below the lane lines and intersecting my path perpendicularly. My interest became piqued. As I swam, I watched the red clocks surrounding the pool and observed that he swam breast-stroke quite regularly, finishing 50 meters (down and back, under the lane-lines) in about 100 seconds.
Ah, I thought. Another math problem. If we both start at the same place (let's say a corner of the pool, just to make this conceptually more straight-forward) and swim in paths perpendicular to each other, when (if at all) will we collide? If we both leave one end of the pool at the same time and swim at our own constant paces, when (if at all) will we collide? If a moment during our infinite-length swim is chosen randomly, what is the probability that we will be in a collision at that moment? Consider that it would be about 7 seconds for me to be in his way; he'd be in my way for 1/9th of the way across the 25 meter pool.
Well, now, TPC students, here's your task. You can either choose to address the probability problems posed in this blog, or you can choose to create your own solvable probability problems that you see in your daily life. Have fun, as always. Be sure your work is original (don't just copy someone else's work) and includes some significant thought. I encourage you to write something meaningful or real to you. Feel free to be poetic or funny.
To answer the question on the most likely inconsiderate lumber man i would say the chances of the man entering the lane right next to you at that exact time are very low. Since there are 16 lanes and you chose a middle lane, there is a 1/8 chance the man would jump in a lane directly next to you. Usually people want to be further away from strangers so i find it strange that the man chose a lane right by you (this decision is probably a factor in the probability but i don't know how to calculate it). Finally, it takes you 45 seconds to swim 50 meters. During 7 of those seconds you are at the side of the pool where the lumpy man is getting ready to jump. So the probability of the man jumping in the pool while you are at the same side of the pool is 7/45. The answer of your question would be 1/8 x 7/45 = 7/360. This probability would be even less if you factored in that he could have gone to the pool at any time of the day. I think the man was trying to be rude based on this data...
ReplyDeleteOne time my brother and i played a fun game with chips. My brother got out 2 sea salt and vinegar kettle cooked chip and 8 normal kettle cooked chips (the chips look the same). I absolutely loathe vinegar and the game was i had to eat 3 out of the 10 chips. It was a really fun game and i did end up having to eat a vinegar chip along with 2 normal chips. The probability that i would get a vinegar chip would be (8C2 x 2C1)/(10C3)= 7/15 probability
You're late to school. For some strange reason you only have time to grab 3 of your schoolbooks. One of these schoolbooks is math, one of these schoolbooks is history, and you don't look at which subject the third book is. If you have a total of 7 textbooks and seven classes but science class doesn't meet that day, what is the probability that the unknown book you grabbed is a Spanish book?
ReplyDeleteAnother probability example:
Your mom asks you to go to the grocery store because she needs milk. You walk to the milk isle, and notice that there are 13 milk jugs in total which you can choose from. Without looking at the labels to see if the milk is spoiled or not, you pull off three jugs, pay for them, and leave. If there were 4 spoiled milk jugs on the shelf and 9 good milk jugs on the shelf, what is the probability that you grabbed at least one spoiled milk jug?
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ReplyDeleteI share your frustration in the pool. During warmup for big meets there are so many kids in a lane that you can barley move. It is head to toe to head to toe. Most kids understand that there is little room or tolerance for varying speeds or strokes. Most of us normal people swim freestyle quietly for the half an hour or so of warmup. But, there is always an exception. There are those few kids on the team who feel the need to sprint during warmup and maul the people in front of them. Unfortunately, all this past weekend I was burdened by these people, during each of the 6 warmup sessions (the meet lasted 3 days, with prelims in the morning and finals at night). Each session I had one of these jerks behind me, scratching my feet and trying to go fast. There were 120 kids from Charger at the meet. 40 of them were in my lane. Out of the 120 there were maybe 10 kids who are “feet touchers”. What were the chances that the there would be one of these kids behind me every session? To find this i would need to know the probability that one of he/she would be in my lane and then the probability that he/she would be directly behind me out of all 40 kids. Then lastly i would need to multiply this by 5 because he/she was there all 5 sessions. This is not a simple problem, but i know the probability is low.
ReplyDeleteA couple of probability examples:
ReplyDelete1) Four light bulbs in a fixture have burned out. From the box in the garage, you take four with which to replace them. However, the box has been there awhile, and six of the 19 in the box don't work. What is the probability that you take 4 defective ones? Two defective ones? No defective ones?
2) I wrote this one, or one similar, in preparation for the last test. Of a group of 50 students, 20 play the piano, 16 play cello, and 14 play flute. 8 students play all 3. 12 play only piano, 10 play only cello, and 8 play only flute.
a) Given that the student plays piano, what is the probability that they play either or both of the other two?
b) Given that they play cello, what is the probability that they do not play either of the others?
c) What is the probability that a student from the group does not play any of the three?
Your neighborhood is having a Easter egg hunt for the kids on Easter sunday. The eggs can be filled with chocolate, jelly beans, or money. There are 600 eggs in total, 200 filled with chocolate, 300 filled with jelly beans, and 100 filled with money. You don't know what is inside. If you find a total of 24 Easter eggs, what is the probability you found 10 chocolate filled and 4 money filled?
ReplyDeleteYou're working at the U.S. post office warehouse. You have to deliver 25 boxes today. If 5 boxes have to go out to the west side, 9 boxes have to go out to the east side, and the other 11, have to go to the valley. You choose 15 of these boxes to deliver first. What is the probably you chose 4 boxes that have to go to the west side, and 1 that has to go to the valley?
If you have 10 pieces of pottery waiting to be fired. There are 4 shelves in the kiln, on which there are 6 spots each. There are 24 pieces that aren't yours waiting to be fired. You are hoping your pieces will get fired, and the more of them that end up on the bottom shelf the happier you are (because it is the hottest place.).
ReplyDeleteIf the kiln is filled at random, what is the probability
that all your pieces will get fired?
that at least one will?
that the bottom shelf will be filled entirely with your pieces?
that at least one of your pieces will end up on the bottom shelf?