Understanding Sample Space and Sample Points

In tossing a fair coin, there are only two possible outcomes, a Head (H) and a Tail (T).  If we let S be the set of all possible outcomes of this event, then, we write the set of possible outcomes as S = {H,T}.

If two fair coins are tossed, then the outcomes can be both heads {H,H} or both tails {T,T}. It can also be a head first then a tail {H,T}, or a tail first and then a head {T,H}. So, in tossing two coins, we have the set of possible outcomes S = {{T,T}, {T,H}, {H,T}, {H,H}}.

As the number of tosses increases, listing gets more difficult. One of the strategies that can be used to remedy this problem is by creating a tree diagram. The following problem is solved using a tree diagram. Notice that it is a three-coin tossing problem in disguise (try replacing B with H and G with T). Continue reading

Wedding Guests and Circular Permutations

In a wedding banquet, guests are seated in circular table for four. In how many ways can the guests be seated?

We have learned that the number of permutations of n distinct objects on a straight line is n!. That is, if we seat the four guests Anna, Barbie, Christian, and Dorcas, on chairs in on a straight line they can be seated in 4 \times 3 \times 2 \times 1 = 24 ways (see complete list).

However, circular arrangement is slightly different. Take the arrangement of guests A, B, C, D as shown in the first figure.  The four possible seating arrangements are just a single permutation: in each table, the persons on the left and on the right of each guest are still the same persons. For example, in any of the tables, B is on the left hand side of A and D is on the right hand side of A. In effect, the  four  linear permutations ABCD, BCDA, CDAB, and DABC are  as one in circular permutation. This means that the number of linear permutations of 4 persons is four times its number of circular permutations.  Since the number of  all possible permutations of four objects is 4!, the number of circular permutations of four objects is \frac{4!}{4}. Continue reading

Lottery Math – If You Play You Should Know Your Odds

Pop Quiz

Here’s a little pop quiz for you, my friends. Sorry, it’s a math quiz. But don’t worry, it’s easy and this is a math blog, after all. Little Johnny, a fifth grader, came home from school one day, visibly upset. His mommy asked what was wrong. Little Johnny told her that he lost out on a treat that his teacher had. He explained to his mommy that his teacher had just one treat, but there were ten students in the class. So each of the students put their names into a hat. Then the teacher randomly chose one name to receive the treat. The rest of the class got nothing. Little Johnny didn’t get the treat. He was upset.

Are you ready for winter?

To comfort him, mommy told Little Johnny that the odds of him getting the treat were against him from the start; his odds were a mere 1-in-___ (Fill in the blank).

Little Johnny didn’t appreciate his mommy spewing out odds as if she was a bookie. Seriously, is that all about, mommy? However, being a simple fifth grade math question, he knew the answer (as should you).

Continue reading

Probability Terminologies and Notations

We had several discussions about probability and before we delve deeper on this topic, let us reinforce our knowledge by familiarizing ourselves with the terminologies and notations used.   This is in preparation to more discussions ahead. Aside from probability, we will also learn more about permutations, combinations, statistics and other related fields. The following are the common terms used in probability as well as the notations used in most textbooks.

Terminologies

Outcome

If a coin is tossed, when the coin comes to rest, it can show a tail or a head, each of which is an outcome.

Trial

Each roll of a die or toss of a coin is a trial.

Experiment

An experiment consists of one or more trials. Continue reading

Experimental and Theoretical Probability Part 3

This is the third part of the Experimental and Theoretical Probability Series.

In the second part of this series, we have observed in three different experiments that if two dice are rolled, it seems that the probability of getting the sums are not equal. Not only that, we have seen several consistent patterns; for example, 2 and 12 got the least number of rolls; while, 6,7, and 8 got the most.

To investigate this observation, we examine how to get a sum of 2, 12, and 6 first when we roll two dice, and then investigate other sums later.  Recall that in the first part of this series, we experimented with two dice, one colored blue and the other red.  To distinguish which number belongs to which dice, we color the numbers blue and red to denote blue and red dice. Continue reading

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