Penny Toss

Date: 12/17/97 at 16:53:39
From: Dave Kugelstadt
Subject: Penny Toss

Dear Dr. Math:

My son is in the 7th grade. I try to help him grasp the concepts of 
advanced math because I believe that a firm understanding of math will 
do a great deal to advance his quality of life, as it has mine.  
Usually I am very good but I am not quite sure about this one.  His 
"Problem of the week 6" goes as follows...

"Three people each toss a penny at the same time.  What is the 
probability that two people get the same side of the penny and the 
other person gets the opposite side?"

Since there was no specification as to which individual got what, I 
reasoned that there are only two possible combinations, all of them 
get the same side or two of them get one side and one gets the other. 
With this I decided that once I determined the odds of all getting the 
same side I could subtract that from 100% to get the chances of any 
two getting the same side. Assuming all else is equal, there is a 1/2 
chance that any one person gets any one particular side. Since there 
are three people I calculated .5 x .5 x .5 = .125 or 12.5% chance that 
all would get the same side. That leaves 87.5% chance that any two 
people would get the same side if all three tossed their pennies at 
the same time (or even not at the same time, I suppose).

Before I tell my son this is it I would like to know if my reasoning 
is sound.

Thanks,
Dave K.

Date: 12/17/97 at 19:25:35
From: Doctor Tom
Subject: Re: Penny Toss

Close, but not quite.

If there are only 3 pennies, it's easy just to list the possibilities:

   HHH
   HHT
   HTH
   HTT
   THH
   THT
   TTH
   TTT

where the first column represents the result for the first person,
etc. So there are 8 equally likely ways the experiment can come out, 
6 of which have two faces the same. Thus the probability is 6/8 = 75%.

You were on the right track, but the .5*.5*.5 is the probability that 
all three throw heads (and also the probability that all three throw 
tails). So there is a 12.5% chance that all three throw heads, a 12.5% 
chance that all throw tails, and hence, a 100% - 12.5% - 12.5% = 75% 
chance that all three flips aren't the same.

I don't know if your kid knows about combinations (like "6 choose 2"
- the number of ways of picking 2 things from a set of 6), but if
so, that's a good way to work it.

There are 3 flips, so there are 2^3 = 8 (2 cubed = 8) possible 
outcomes, and the favorable outcomes are if there is 1 head of the 
three or 2 heads. (If there are 0 or 3 heads, it's an unfavorable 
outcome.)  The number of favorable outcomes is thus

(3 choose 1) + (3 choose 2) = 3 + 3 = 6 favorable out of 8 possible,
or a probability of 6/8, or 75%.

This last method is far more powerful in terms of answering more
complex problems, but you have to know how to count combinations.

For example, if the problem were, "7 people flip 1 penny each. What is 
the probability that there are at least 4 heads tossed?"

Well, there are 2^7 = 128 outcomes, and there are:

(7 choose 4) + (7 choose 5) + (7 choose 6)  (7 choose 7)

favorable outcomes.

If you work it out, this is 35+21+7+1 = 64 favorable of 128, giving
a probability of 64/128, or 1/2.

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 12/17/97 at 19:28:16
From: Doctor Anthony
Subject: Re: Penny Toss

You are partly but not completely right.

The easiest way is to consider the probability of NOT getting two of 
one and one of the other.  This would be either HHH or TTT

The chance of HHH is (1/2)^3   and of TTT is also (1/2)^3

                  = 1/8 + 1/8 = 1/4

So the chance of two of one and one of the other is  3/4

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/