Return to Free Tour Crazy Animals
Content Epilogue

 

How Many Possible Animals Are There?

Consider just two animals (horse and giraffe) and three parts for each (head, body and legs).

Tree Diagram A tree diagram shows the various combinations - eight pathways to eight end points.

Alternatively, with two choices at each branch, the total of eight could be obtained by multiplying 2x2x2.

So for three animals, there will be three options at each branch, giving a total of

3x3x3 = 27
Try drawing the tree diagram to confirm this.

Probability of Making a Complete Animal

Given eight different combinations, the chance of a complete giraffe being made is one chance out of eight. That is, we would expect one student out of every eight to create a complete giraffe. If there are 16 students in the class, we would expect two giraffes. If we had 24 students, we would expect three giraffes. If we had 200 people we would expect 25 giraffes. (This can be checked with the software using Option 4.)

200 part animals

Notice that although the actual number of giraffes is close to what is expected, it is not exactly what is expected. See Natural Variability below.

With three animals and 27 different combinations, the chance of getting a complete giraffe is 1 in 27. So, if we had 27 children in the group we would expect one complete giraffe (and also one horse and one duck).

Natural Variability

The theoretical probability described above gives a long-term prediction, that is, it tells us the average fraction of a very large number of crazy animals that will be one particular result. For relatively small numbers of rolls of the dice, like the single set of 200 in the picture above, this fraction cannot be expected to give the outcome.

The Multiplication and Addition Laws

This investigation gives examples of these 'laws' in action. The laws are simply common sense rules that enable complex probabilities to be worked out from simpler ones.

Multiplication
Three different rolls of a dice are used to decide whether each of the head, body and legs are from a giraffe or a horse. To find the probability of getting a complete giraffe we can multiply the fractions for the probabilities of getting a giraffe head, a giraffe body and a giraffe legs. So, the probability of getting a complete giraffe is:

1/2 x 1/2 x 1/2

Addition
To get the probability of all the two-part giraffe animals we can add the probability of getting each one. Since each crazy animal has the same probability of being created, 1/8, and there are three crazy animals which are two-parts giraffe, the probability of getting a two-part giraffe is 3/8.

These laws are really just common sense formalised.

 

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