## What are Equally Likely Outcomes?

Equally likely outcomes are, like the name suggests, events with an equal chance of happening. Many events have equally likely outcomes, like tossing a coin (50% probability of heads; 50% probability of tails) or a die (1/6 probability of getting any number on the die).

In real life though, **it’s highly unusual to get equally likely outcomes for events**. For example, the probability of finding a golden ticket in a chocolate bar might be 5%, but this doesn’t contradict the idea of equally likely outcomes. Let’s say there are 100 chocolate bars and five of them have golden ticket, which gives us our 5% probability. Each of those golden tickets represents one chance to win, and there are five chances to win, each of which are equally likely outcomes. Other examples:

- Flip a fair coin 10 times to see how many heads or tails you get. Each event (getting a heads or getting a tails) is equally likely).
- Roll a die 3 times and note the sequence of numbers. Each sequence of numbers (123,234,456,…) is equally likely.

## How to Find the Probability of Equally Likely Outcomes

Formally, equally likely outcomes are defined as follows:

For any sample space with N equally likely outcomes, we assign the probability 1/N to each outcome.[1]

To find the probability of equally likely outcomes:

- Define the sample space for an event of chance. The sample space is all distinct outcomes. For example, if 100 lottery tickets are sold numbered 1 through 100, the sample space is a list of all winning tickets (1, 2, 3, …, 100).
- Count the number of ways event A can occur. For this example, let’s say that event A is “picking the number 33”. There is only one way to choose the number 33 from the list of numbers 1 through 100.
- Divide your answer from (2) by your answer from (1), giving: 1/100 or 1%.

A slightly more complicated example. Let’s say you were interested in calculating the probability of choosing any ticket with the number three.

- The sample space is still a list of all winning tickets (1, 2, 3, …, 100).
- Event A, “picking a ticket with the number 3”, has ten possibilities: 3, 13, 23, 33, 43, 53, 63, 73, 83, 93.
- Divide your answer from (2) by your answer from (1), giving: 10/100 = 10%.

## References

[1] Equally Likely outcomes. Retrieved February 19, 2021 from:

https://www3.nd.edu/~dgalvin1/10120/10120_S16/Topic09_7p2_Galvin.pdf

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