math question

[demoskopico]

Hello, I don't know if it is the right place to write down this quant quaestion..anyway I'll do that so that I could find somebody able to explain that to me...

The figure shown represents a board with 4 rows of pegs, and at the bottom of the board are 4 cells numbered 1 to 4. Whenever the ball shown passes through the opening between two adjacent pegs in the same row, it will hit the peg directly beneath the opening. The ball then has the probability 1/2 of passing through the opening immediately to the left of that peg and probability 1/2 of passing through the opening immediately to the right. What is the probability that when the ball passes through the first two pegs at the top it will end in Cell 2?


A. 1/16
B. 1/8
C. 1/4
D. 3/8
E. 1/2

see file attached.

thanks a lot

[JohnB]

Hi Demoskipo,

Good question--although the sketch didn't quite come out, I think I see how this is supposed to work. The ball drops between the top two pegs and hits the peg in the middle of row 2. To figure out the probability for its final location, we should look at the possible routes it could travel from row 2. There are 8 possibilities, with L meaning the ball goes left, and R meaning it goes right:

It could go LLL--this puts it into cell 1.
It could go LLR--this puts it into cell 2.
It could go LRL--this puts it into cell 2.
It could go LRR--this puts it into cell 3.
It could go RLL--this puts it into cell 2.
It could go RLR--this puts it into cell 3.
It could go RRL--this puts it into cell 3.
Or it could go RRR--this puts it into cell 4.

There are 8 total possibilities, and 3 of them give us a result of cell 2. So the probability of cell 2 is 3/8.

All best,
John

[demoskopico]

Hi Demoskipo,

Good question--although the sketch didn't quite come out, I think I see how this is supposed to work. The ball drops between the top two pegs and hits the peg in the middle of row 2. To figure out the probability for its final location, we should look at the possible routes it could travel from row 2. There are 8 possibilities, with L meaning the ball goes left, and R meaning it goes right:

It could go LLL--this puts it into cell 1.
It could go LLR--this puts it into cell 2.
It could go LRL--this puts it into cell 2.
It could go LRR--this puts it into cell 3.
It could go RLL--this puts it into cell 2.
It could go RLR--this puts it into cell 3.
It could go RRL--this puts it into cell 3.
Or it could go RRR--this puts it into cell 4.

There are 8 total possibilities, and 3 of them give us a result of cell 2. So the probability of cell 2 is 3/8.

All best,
John

Hi..I really don't catch your explanation but you're right! OA is 3/8....i don't understand the sense....please help me!

[JohnB]

Hi Demoskipo,

Here's a more detailed account of how I'm approaching this:

a) We're trying to find the probability of landing in cell 2. In general, we can define probability of a certain event as the ratio of the number of ways that event can happen to the total number of possibilities.

b) Using this definition, I want to find both the total number of possibilities, and the number of ways that the ball can drop into cell 2.

c) I can see based on the diagram that the ball will drop onto the middle peg of row 2. It then has two more rows to go through until it reaches the bottom row of cells. In each row, it has a 50% chance of moving to the left, and a 50% chance of moving to the right.

d) Since there are 3 rows that the ball must pass through (I ignore the first row, since the ball drops between its two pegs), and in each row it can move in one of two directions, the total number of possibilities for paths that the ball can travel is 2 to the third power, or 8.

e) In my earlier post, I set out all the potential ways that the ball can travel down the board. Three of them lead to it falling into cell #2. So there are 3 ways the event we want can happen, and 8 total possibilities. The probability overall is 3/8.

Let me know if you have further questions. In particular, let me know specifically if there are steps above that don't make sense to you.

All best,
John

[demoskopico]

Hi Demoskipo,

Here's a more detailed account of how I'm approaching this:

a) We're trying to find the probability of landing in cell 2. In general, we can define probability of a certain event as the ratio of the number of ways that event can happen to the total number of possibilities.

b) Using this definition, I want to find both the total number of possibilities, and the number of ways that the ball can drop into cell 2.

c) I can see based on the diagram that the ball will drop onto the middle peg of row 2. It then has two more rows to go through until it reaches the bottom row of cells. In each row, it has a 50% chance of moving to the left, and a 50% chance of moving to the right.

d) Since there are 3 rows that the ball must pass through (I ignore the first row, since the ball drops between its two pegs), and in each row it can move in one of two directions, the total number of possibilities for paths that the ball can travel is 2 to the third power, or 8.

e) In my earlier post, I set out all the potential ways that the ball can travel down the board. Three of them lead to it falling into cell #2. So there are 3 ways the event we want can happen, and 8 total possibilities. The probability overall is 3/8.

Let me know if you have further questions. In particular, let me know specifically if there are steps above that don't make sense to you.

All best,
John

each thing is clear but one: why there are three ways the event can heppen?
i don't catch the patterns you draw before...i.e. LLL, LLR...maybe because my diagram is not clear. I would be very happy if you, if it doesn't bother you, draw each pattern on the graph so that I can distinguish it...maybe I find difficulties because I am not native english and the argument is not so easy...

thanx once again!

[kevin]

Marco. can you see the similarity between the question you posted and the following?

Marco and Maria toss a coin three times. Each time a head is obtained, Marco gives Maria $1. Each time a tail is obtained, Maria gives Marco $1. What is the probability that after the 3 tosses, Marco has $1 less than he did before the 3 tosses?