How-Tos / Math

How to Find Probability

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How to Find Probability

Last updated on September 25th, 2020

Probability is the likelihood of one or more events happening divided by the number of possible outcomes. You can break down the problem into separate probabilities and multiple the separate likelihoods by one another to count the probability of multiple events. Here is how to calculate the probability of single and multiple random events and also converting odds to probability. 

\[Probability = {The\ number\ of\ wanted\ outcomes \over The\ number\ of\ possible\ outcomes}\].

How to Find the Probability of a Single Random Event

  1. Determine a single event with a single outcome.

    Example: The probability to get a 6 when you roll a die.

    A die has 6 sides, 1 side contains the number 6 that give us 1 wanted outcome. 

  2. Identify the total number of outcomes that can occur.

    There are six possible outcomes when you roll a die. 

  3. Divide the number of events by the number of possible outcomes.

    Probability = {The\ number\ of\ wanted\ outcomes \over The\ number\ of\ possible\ outcomes} = {1 \over 6}

  4. Add up all possible event likelihoods to make sure they equal 1.

    The likelihood of rolling a 6 on a 6-sided die is 1/6. But the probability of rolling all five other numbers on a die is also 1/6. 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 6/6 , which = 100%.

How to Find the Probability of a Multiple Random Events

There are four kinds of multiple random events and which are Independent Events, Dependent events, Exclusive events, and Inclusive events. 

Independent events: Two events are independent when the outcome of the first event does not influence the outcome of the second event. Multiply the probability of the first event by the probability of the second event is the probability of two independent events. 

P(X and Y) = P(X) ⋅ P(Y)

Example: If one has three dice what is the probability of getting three 4s? 

  1. Determine each event you will calculate.

    The probability of getting a 4 on one die is 1/6

  2. Calculate the probability of each event.

    The probability of getting 3 4s is: 1/6, 1/6, 1/6

  3. Multiply all probabilities together.

    P (4\ and\ 4\ and\ 4) = \frac{1}{6}.\ \frac{1}{6}.\ \frac{1}{6} = \frac{1}{216}

Dependent Events: Two events are dependent when the outcome of the first event influences the outcome of the second event. The probability of two dependent events is the product of the probability of X and the probability of Y AFTER X occurs.

P(X and Y) = P(X) ⋅ P(Y after X)

Example: What is the probability for you to choose two red cards in a deck of cards?

  1. Determine each event you will calculate.

    A deck of cards has 26 black and 26 red cards. The probability of choosing a red card randomly is:

    P(red) = \frac{26}{52} = \frac{1}{2}

  2. Consider the effect of prior events when calculating the probability of dependent events.

    The probability of choosing a second red card from the deck is now:

    P(2\ red) = \frac{25}{51}

  3. Multiply all probabilities together.

    The probability:  P(2\ red) = \frac{1}{2}\cdot \frac{25}{51} = \frac{25}{102}

Exclusive Events: Two events are mutually exclusive when two events cannot happen at the same time. The probability that one of the mutually exclusive events occur is the sum of their individual probabilities. 

P(X or Y) = P(X) + P(Y)

Example: Ace OR King

In a Deck of 52 Cards:

  • the probability of an Ace is 1/13, so P(Ace)=1/13
  • the probability of a King is also 1/13, so P(King)=1/13

When we combine those two Events:

  • The probability of a King or a Queen is (1/13) + (1/13) = 2/13

Which is written like this: P(Ace or King) = (1/13) + (1/13) = 2/13

Inclusive events are events that can happen at the same time. To find the probability of an inclusive event we first add the probabilities of the individual events and then subtract the probability of the two events happening at the same time.

P(X or Y) = P(X) + P(Y) – P(X and Y)

Example: The probability of drawing a black card or a ten in a deck of cards.

There are 4 tens in a deck of cards P(10) = 4/52

There are 26 black cards P(black) = 26/52

There are 2 black tens P(black and 10) = 2/52

P (black\ or\ ten) = \frac{4}{52} + \frac{26}{52} - \frac{2}{52} = \frac{30}{52} - \frac{2}{52} = \frac{28}{52} = \frac{7}{13}

How to Convert Odds to Probability

Determining the likelihood of this event actually occurring is referred to as the odds. The odds, or chance, of something happening, depends on the probability. The odds take the probability of an event occurring and divide it by the probability of the event not occurring.

Example: Colored Marbles. Figure out the probability of drawing a white marble (of which there are 11) out of the total pot of marbles (which contains 20). 

  1. Set the odds as a ratio with a positive outcome as a numerator.

    There are 11 white and 9 non-white marbles, you’ll write the odds as the ratio 11:9

  2. Add the numbers together to convert the odds to probability.

    The event that you’ll draw a white marble is 11; the event another color will be drawn is 9. The total number of outcomes is 11 + 9, or 20.

  3. Find the odds as if you were calculating the probability of a single event.

    The probability of drawing a white marble is 11/20. Divide this out: 11 ÷ 20 = 0.55 or 55%