Three Coins Three coins are tossed.?

Three coins are tossed.

? a) Determine the number of points in the sample space = 2^3 = 8

? b) Construct a tree diagram and list the sample space. YOU DO IT.

Determine the probability that

? c) no heads are tossed P(no heads) = P(all tails) = (1/2)^3 = 1/8

? d) exactly one head is tossed. P(exactly one head) = 3(1/2)(1/2)^2 = 3/8

? e) three heads are tossed. P(three heads) = (1/2)^3 = 1/8

Three Coins Three coins are tossed.

? a) Determine the number of points in the sample space.

The sample space is the list of all possible outcomes:

HHH, HHT, HTH, HTT,THH, THT, TTH, TTT

where H=heads, T=tails.

There are 2 possible outcomes, H or T, for each coin, and the outcome for one coin is independent of the outcomes of the other two coins.

So size of the sample space is 2 x 2 x 2 = 23 = 8.

Another way to look at this is to think of the possible outcomes for each coin as 0 or 1. Count in binary from 000 to 111 to get the size of the sample space.

? b) Construct a tree diagram and list the sample space.

Here are some examples of tree diagrams:

http://www.google.com/images?q=tree+diag…

http://regentsprep.org/Regents/math/ALGE…

Determine the probability that

? c) no heads are tossed.

The probability P of an event is the number of ways the event can occur divided by the number of all possible outcomes.

Only way for this event to occur: TTT.

So P = 1/8.

? d) exactly one head is tossed.

There are three ways for this to event to occur: HTT, THT, and TTH

So P = 3/8.

? e) three heads are tossed.

Only one way for this event to occur: HHH

So P = 1/8.

A coin has only 2 sides, so for each flip there are only two possible results.

a) Determine the number of "points" in the sample space.

The sample space is all the possible results of the 3 tosses taken together, so:

We could have 3 heads in a row, denoted as H H H

Or,

H H T

H T H

T H H

Yes, even though the one Tail may occur in a different toss, it still counts as a different result in the sample space. Continuing...

T H T

H T T

T T H

T T T

There are 8 total results of tossing 3 coins in a row. And the total results is the number of points, or the size, of the sample space: 8.

b) Tree diagram:

The way these are made is that you start the tree off at the beginning, at the first toss, and it branches for each following event:

The first toss The second toss The third toss

____________________H

/

_____________________H

/ \____________________T

/

_________H ____________________H

/ \ /

/ \______________________T

/ \____________________T

/ ________________________H

\ /

\ _____________________H

\ / \________________________T

\ _________T ____________________H

\ /

\______________________T

\_____________________T

This tree is used to "map" the possible outcomes on a series of events. Basically, tossing 3 coins would follow a path from the 'root' down the tree's branches to the last 'leaf'. Also note that the number of final branches (also called leaves) is the same number from a, the number of points in the sample space.

c) Determine the probability that no heads are tossed. If no heads are tossed, the tosses must all be tails, so that will look like T T T. How many different sets of 3 tosses result in all tails? Only one! You can even use the tree to find out how many paths from root to leaf there are where no heads are encountered. Again the answer is 1.

d) Determine the probability that exactly one head is tossed. Well now, this ONE head could be on the first toss, the second, or the third, but the other two tosses cannot be heads, so they must be tails. Again we can use the tree to see how many different paths only have one head (H) in them. The answer is 3, there are only 3 different possible outcomes of tossing three coins where you only get a head once: T H T, H T T, and T T H. Try as you may, you will find that no other sets of 3 tosses have only one head, they have 2, 3, or none.

e) Determine the probability that three heads are tossed. Well this is similar to c. If we have all heads, we must have no tails. There is only one possible set of 3 where there are all heads: H H H. The answer here is 1.

The number of points is basically the number of possibilities to the power of how many trials. In this case, there are 2 possibilities, and 3 tosses, so 23 = 8

Number of points = 8

Here's a tree diagram of this situation

http://cnx.org/content/m19234/latest/gra…

And here is the sample space

S = {(HHH) , (HHT) , (HTH) , (HTT) , (THH) , (THT) , (TTH) , (TTT)}

No heads are tossed:

This is the same as saying the chances of 3 tails. The chances of getting tails each time is 1/2 so multiply the probabilities together 3 times

(1/2)(1/2)(1/2) = (1/2)(1/4) = 1/8

Exactly one head:

There are 3 possibilities that give exactly 1 head, so the probability is 3/8

3 heads:

There is 1 possibility of getting 3 heads, so the probability is 1/8