4.2 Introduction to Computing the Probability of an Event

Learning to Calculate the Probability of Events

Probability is the likelihood of a particular outcome or event happening. Statisticians and actuaries use probability to make predictions about events.  An actuary that works for a car insurance company would, for example, be interested in how likely a 17-year-old male would be to get in a car accident. They would use data from past events to make predictions about future events using the characteristics of probabilities, then use this information to calculate an insurance rate.

Probabilities are between zero and one, inclusive (that is, zero and one and all numbers between these values).

  • P(A) = 0 means the event A can never happen.
  • P(A) = 1 means the event A always happens.
  • P(A) = 0.5 means the event A is equally likely to occur or not to occur.

Equally likely means that each outcome of an experiment occurs with equal probability. For example, if you toss a fair, six-sided die, each face (1, 2, 3, 4, 5, or 6) is as likely to occur as any other face. If you toss a fair coin, a Head (H) and a Tail (T) are equally likely to occur. If you randomly guess the answer to a true/false question on an exam, you are equally likely to select a correct answer or an incorrect answer.

To calculate the probability of an event A when all outcomes in the sample space are equally likely, count the number of outcomes for event A and divide by the total number of outcomes in the sample space. For example, if you toss a fair dime and a fair nickel, the sample space is {HH, TH, HT, TT} where T = tails and H = heads. The sample space has four outcomes. A = getting one head. There are two outcomes that meet this condition {HT, TH}, so P(A) = 2/4 = 0.5.

Suppose you roll one fair six-sided die, with the numbers {1, 2, 3, 4, 5, 6} on its faces. Let event E = rolling a number that is at least five. There are two outcomes {5, 6}. P(E)=2/6.  This simplifies to P(E) = 1/3

This important characteristic of probability experiments is known as the law of large numbers which states that as the number of repetitions of an experiment is increased, the empirical probability of the experiment tends to become closer and closer to the theoretical probability. Even though the outcomes do not happen according to any set pattern or order, overall, the long-term empirical probability will approach the theoretical probability.

Basic Probability

Given that all outcomes are equally likely, we can compute the probability of an event E using this formula:

[latex]P(E)=\frac{\text{Number of favorable outcomes corresponding to the event E}}{\text{Number of equally-likely outcomes (sample space)}}[/latex]

examples

If we roll a 6-sided die, calculate

  1. P(rolling a 1)
  2. P(rolling a number bigger than 4)

This video describes this example and the previous one in detail.

Let’s say you have a bag with 20 cherries, 14 sweet and 6 sour. If you pick a cherry at random, what is the probability that it will be sweet?

Cards

A standard deck of 52 playing cards consists of four suits (hearts, spades, diamonds and clubs). Spades and clubs are black while hearts and diamonds are red. Each suit contains 13 cards, each of a different rank: an Ace (which in many games functions as both a low card and a high card), cards numbered 2 through 10, a Jack, a Queen and a King.

example

Compute the probability of randomly drawing one card from a deck and getting an Ace.

This video demonstrates both this example and the previous cherry example on the page.  Note that sometimes a probability is interpreted as a percent.  In our class we will give the probability as a simplified fraction or a number rounded to 3 decimal digits.  (i.e. 0.729)

Try It

 

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