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Lesson 7. Probability

Computing probability is simple, it is: SPECIFIC / TOTAL. What does it mean? Let's start with an example.

Example. Suppose that you have a (six-sided) regular dice, whose sides contain numbers $1, 2, 3, 4, 5, 6$. If you throw the dice, the probability that you will get 6 can be computed as follows:
  • TOTAL ($1, 2, 3, 4, 5, 6$) = $6$ sides;
  • SPECIFIC (only $6$) = $1$ sides.
Therefore, the probability that you get $6$ from throwing the dice once is equal to $1/6$. 

Exercise. What is the probability that you get $3$ by throwing the same dice?

Solution. TOTAL = $6$, as there are six sides for the dice, and SPECIFIC = $1$, because there is one side that has $3$. Thus, the probability that you get $3$ from throwing the dices once is equal to $1/6$. This is the answer.

Let's spice up our example a little bit.

Example. Suppose that you have two regular dices, each of which has six sides containing $1, 2, 3, 4, 5, 6$. If you throw both of them at once, what is the probability that the sum of numbers is equal to $10$?

Again, probability is SPECIFIC/TOTAL, and usually it is easier to find TOTAL. What is TOTAL? It is all possible ways to have (first, second), where "first" means the number from the first dice and "second" means the number from the second dice. We may count them as follows:
  1. $(1, 1)$,
  2. $(1, 2)$,
  3. $(1, 3)$,
  4. $(1, 4)$,
  5. $(1, 5)$,
  6. $(1, 6)$, ...
and there are many more. How do we count all of them? Well, note that we have counted all the cases where FIRST = 1, and there are 6 such cases. Now, the same situation will happen when FIRST = 2, 3, 4, 5, 6, so the total number we count is $6 \times 6 = 36$. Hence, TOTAL = $36$.

What about SPECIFIC? We need to count pairs whose numbers add up to $10$. We notice that
  1. $1 + 9 = 10$,
  2. $2 + 8 = 10$,
  3. $3 + 7 = 10$,
  4. $4 + 6 = 10$,
  5. $5 + 5 = 10$,
  6. $6 + 4 = 10$,
  7. $7 + 3 = 10$,
  8. $8 + 2 = 10$,
  9. $9 + 1 = 10$,
but the choices 1, 2, 3, 7, 8, 9 are not valid because each dice only has $1, 2, 3, 4, 5, 6$, while such choices used $7, 8, 9$. Thus, the choices 4, 5, 6 are the only options, meaning SPECIFIC = 3. Therefore, the probability that you will get 10 when you throw both dices at once is equal to $3/36 = 1/12$. This is the answer.

Remark. There are definitely more materials on probability than what we have discussed in this posting for GMAT Quantitative Reasoning. However, we go one step at a time! Try to absorb the above examples, and let me know if you find anything unclear!

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