What is the probability of rolling a six? Of pulling out a red ball? Let’s learn to calculate chances like mathematicians.
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What is probability?
Probability is a number from 0 to 1 that shows how likely something is to happen.
Formula
\(P = \dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}\)
What the extreme values mean
\(P = 0\) — the event is impossible (rolling a 7 on an ordinary die).
\(P = 1\) — the event is certain (you will roll a number from 1 to 6).
\(P = \dfrac{1}{2}\) — equally likely to happen or not.
Example — coin toss
We toss a coin. What is the probability of getting heads?
Total outcomes: 2 (heads or tails).
Favourable outcomes: 1 (heads).
\(P = \dfrac{1}{2}\).
Try it
A bag contains 3 red and 7 blue balls. The probability of drawing a red ball is:
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Rolling a die
An ordinary die has 6 faces with numbers 1, 2, 3, 4, 5, 6. Each number is equally likely.
Typical questions
Probability of an even number: 2, 4, 6 → \(P = \dfrac{3}{6} = \dfrac{1}{2}\).
Probability of rolling a number greater than 4: numbers 5 and 6 → \(P = \dfrac{2}{6} = \dfrac{1}{3}\).
Probability of a prime number: 2, 3, 5 → \(P = \dfrac{3}{6} = \dfrac{1}{2}\).
Try it
We roll a die. The probability that the result is divisible by 3 is:
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Balls in a bag
This is the most popular CMS type. Usually there is a bag with balls of different colours, and you need the probability of drawing a ball of the required colour.
Example
There are 3 red and 5 blue balls in a bag. We draw one at random.
Probability of red: \(\dfrac{3}{3+5} = \dfrac{3}{8}\).
Probability of blue: \(\dfrac{5}{8}\).
Careful: without replacement
If you drew a ball and did not put it did not return it to the bag, the total number decreases!
Originally: 3 red + 5 blue = 8. After drawing one red → 2 red + 5 blue remain = 7.
Probability of a second red: \(\dfrac{2}{7}\), not \(\dfrac{2}{8}\)!
Try it
A bag contains 4 green, 3 yellow, and 5 white balls. The probability of drawing a yellow ball is:
Try it
A bag contains 6 red and 4 white balls. One red ball is drawn and not replaced. The probability that the second one is also red is:
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Probability of “NOT”
Complementary event
\(P(\text{NOT } A) = 1 - P(A)\)
The probability that an event does NOT happen equals 1 minus the probability that it does happen.
This is a very powerful trick! Sometimes it is easier to calculate what we do not wantand subtract from 1.
Example
We roll a die. What is the probability of NOT getting a six?
Three numbers are chosen at random from the set \(\{1, 2, 3, 4, 5, 6\}\). How many ways are there to choose three numbers with an even sum?
Even: {2, 4, 6}. Odd: {1, 3, 5}.
The sum is even if all three are even, or one is even and two are odd.
All three even: \(\binom{3}{3} = 1\) way.
One even + two odd: \(\binom{3}{1} \times \binom{3}{2} = 3 \times 3 = 9\) ways.
Total: \(1 + 9 = 10\).
(This is CMS 2021 Q24!)
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Cheat sheet — Probability
Basic formula: \(P = \dfrac{\text{favourable}}{\text{total}}\) Complementary event: \(P(\text{NOT }A) = 1 - P(A)\) “AND” (both): multiply the probabilities “OR” (at least one): add the probabilities Without replacement: after each draw, reduce both the numerator and the denominator Die: 6 outcomes, each with probability \(\dfrac{1}{6}\) Two dice: 36 outcomes