Problems on Probability

Two questions on probability that I do not understand. Usually when I read the solutions to the questions, I would be able to understand where the answers are coming from but these two have been bugging me because it doesn't make any sense to me.

Q1: 6 coins are thrown at once.
Solution:

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Q2:


Solution:



Question 1(b), why do we divide the answer in 1(a) with 2⁶?

For question 2, I can't figure out why we're going to divide the number of combinations of taking 2 from 7 horizontal lines with the number of combinations of taking 2 from 12 straight lines. Why not divide it only by 12 instead, since that's total number of the lines on the plane anyway.

Thanks! Probability has been driving me nuts for the past couple of days. It's really difficult.

I feel for you. Took me a long time to wrap my mind around some of the concepts and I still have to be careful or I make a mistake. Both your questions revolve around the definition of probability of something and that is, the probability of x happening, P(x), is defined by
P(x)=\(\displaystyle \frac{The\, number\, of\, ways\, x\, can\, happen}{The\, number\, of\, ways\, everything\, can\, happen}\)
where, of course, everything is everything your are concerned with, i.e. the number of combination of heads and tails possible if you toss six coins in the air.

Cafelatte said:

...
Question 1(b), why do we divide the answer in 1(a) with 2⁶?

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As for dividing by 2⁶, that is the number of possible outcomes. Whether you do the coin toss one a time to get your number of possible outcomes or toss them all at once, the answer is the same. The reason this is true is that each coin toss when you toss them individually is independent of all the other coin tosses.

Cafelatte said:

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For question 2, I can't figure out why we're going to divide the number of combinations of taking 2 from 7 horizontal lines with the number of combinations of taking 2 from 12 straight lines. Why not divide it only by 12 instead, since that's total number of the lines on the plane anyway. ...

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Again, if you pick two lines from 12 lines, the total number of possibilities is C₂¹². To do a manageable number as an example, just pick 2 out of 4 (and label them 1, 2, 3, and 4): 1,2; 1,3; 1,4; 2,3; 2,4; 3,4.

I hope that helps.

Cafelatte said:

Question 1(b), why do we divide the answer in 1(a) with 2⁶?
For question 2, I can't figure out why we're going to divide the number of combinations of taking 2 from 7 horizontal lines with the number of combinations of taking 2 from 12 straight lines. Why not divide it only by 12 instead, since that's total number of the lines on the plane anyway. Thanks! Probability has been driving me nuts for the past couple of days. It's really difficult.

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Frankly, I cannot understand the confusion in the OP nor the first reply. If it hard to understand, I wonder?

It should be very easy to see that there \(\displaystyle \dbinom{12}{2}=\dfrac{12\cdot 11}{2}=66\) ways to pick two lines from twelve. There are \(\displaystyle 5\cdot 7=35\) ways to pick one vertical line and one horizontal line. Those will not be parallel. Therefore there are \(\displaystyle 66-35=31\) ways to pick parallel pairs.

pka said:

Frankly, I cannot understand the confusion in the OP nor the first reply. If it hard to understand, I wonder?
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The statement in the first reply was in reference to some aspects of statistics, not to this particular problem. Had I been confused about this problem, I would not have attempted to alleviate some of the confusion of the poster. Never the less, there are still questions in sadistics I need to be very careful in trying to solve and even then, I am not always successful.

Ishuda said:

The statement in the first reply was in reference to some aspects of statistics, not to this particular problem. Had I been confused about this problem, I would not have attempted to alleviate some of the confusion of the poster. Never the less, there are still questions in sadistics I need to be very careful in trying to solve and even then, I am not always successful.

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Thank you for illustrating my concern. The OP had absolutely nothing to do with statistics. These are purely probability questions. I still have to wonder about not knowing the difference?

pka said:

Thank you for illustrating my concern. The OP had absolutely nothing to do with statistics. These are purely probability questions. I still have to wonder about not knowing the difference?

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OK, so I mispoke. Thank you for pointing that out. However the same statement applies if you make the proper substitution.