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3.1 From Description

Probability, in simple language, is the likelihood of something happening and in mathematical terms is:

Probability (Event A) = (# of ways A can happen) / (Total # of outcomes)

Everything in probability circulates around this one simple concept. (*Side note: There are a few things that have been absolutely favourite in probability questions like dice roll, drawing from card, flipping of coin. Make sure you are confident with the basics of how these work*)

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*Example 1*

Calculate the following probabilities:

- Getting head on a coin toss
- Getting a spade when drawing one card from deck
- Getting an even number on a dice roll
- Selecting a female from a class of 20 males and 30 females

**Probability Properties**

There are few important properties of probability that you should be aware:

- Sum of all probabilities is equal to 1
- Probability of Event A = 1 – Probability of Event A not happening

*Example 2*

Calculate the following probabilities:

- Probability of getting head on a biased coin where probability of tail is 1/3
- Probability of not getting a heart when a card is taken out from standard deck

**Multiple Draws**

Probability questions can get tricky with the introduction of multiple draws. Instead of rolling dice once, drawing it twice or three times etc. Similarly, instead of taking one card out from deck, taking out two or three cards. Note in the first case where dice is rolled the probability of an outcome (e.g. seeing a 3) on a dice roll does not change no matter how many times you roll. In the latter case, the probability will change if the card is drawn out and not put back into the deck (called ‘without replacement’).

*Example 3*

Calculate the following probabilities:

- Probability of getting head on 10
^{th}coin toss - Probability of getting two 3’s on two dice rolls
- Probability of getting two spades and one heart in three draws from standard card without replacement

**Multiple Draws (more complexity)**

With multiple draws comes more complexity. Make sure you know how to solve when question asks you ‘At Least’, ‘Greater than’, ‘Less than’, ‘At Most’.

*Example 4*

Calculate the following probabilities:

- Probability of at least one spade when two cards are taken out from standard deck without replacement
- Probability of more than 2 girls selected when 4 students are selected randomly without replacement from a class of 25 boys and 25 girls
- Probability of less than 3 hearts when three cards are taken out from standard deck without replacement
- Probability of at most one diamond when three cards are taken out from standard deck without replacement

**Financial Expectation**

Many times, these probability questions are linked with money and you need to calculate the expected value. Think of these questions as two-layered questions where one layer asks you to calculate probability of an outcome (or gives you that probability outright in the question) and the other layer assigns a value to that outcome. Expected value then is simply multiplication and sum of each individual probability with the financial value.

*Example 5*

Calculate the following probabilities:

- You win $1 when you get 2 on a dice roll and lose $0.50 on every other outcome. What is the expected value?

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