Probability of Independent Events: The 'At Least One' Rule

Probability of Independent Events: The 'At Least One' Rule

Occasionally when calculating independent events, it is only important that the event happens once. This is referred to as the 'At Least One' Rule. To calculate this type of problem, we will use the process of complementary events to find the probability of our event occurring at least once.

Independent Events

Independent events are events that do not affect the outcome of subsequent events. In an independent event, each situation is separate from previous events. An example of independent events would be the probability that it will rain on Monday, and the probability of getting an A on my next test. These two events are independent of each other. The chances that it will rain on Monday does not affect the score on my next test. To calculate the probability of multiple independent events, find the probability of each event happening separately and multiply them together.

'At Least One Rule'

Occasionally when calculating independent events, it is only important that the event occurs at least once. This is referred to as the 'At Least One' Rule. To calculate the probability of an event occurring at least once, it will be the complement of the event never occurring. This means that the probability of the event never occurring and the probability of the event occurring at least once will equal one, or a 100% chance.

For example, the probability of winning the grand prize in a local drawing is 1 out of 30. Tim and his wife, Jane, both bought tickets. What is the probability that at least one of them will win the grand prize? We first need to find the probability of Tim and Jane not winning the grand prize. Since the probability of one person winning the prize is 1 out of 30, the probability of one person not winning the grand prize is 29/30, or 0.96.

Remember, to calculate the probability of multiple independent events - in this case, both of them not winning the grand prize - we find the probability of each event happening separately and multiply them together. Since both Tim and Jane have the same probability of not winning, we will need to square the probability of one not winning. So, 0.967^2 = 0.935.

So, the probability that neither Tim nor Jane will win the grand prize is 0.935. To calculate the probability of at least one of them winning the grand prize, we need to find the complement of that number. The probability of not winning plus the probability of at least one winning is going to equal one whole. So, by subtracting 1 - 0.935, we can see that the probability of either Tim or Jane winning the grand prize is 0.065, or a 6.5% chance.

As the announcer steps to the mic to call out the winning ticket, Tim and Jane do not like their chances. The announcer steps to the mic and calls out the name Jane. Jane is so excited and jumps with joy that she has beaten the odds to win the grand prize.

Example

Let's look at another example. Tim and Jane are planning to use their grand prize winnings to take a four-day ski trip. They want to make sure that it will snow at least one day while they're on their trip. The ski resort that they booked with says that there is a 65% chance that it will snow each day. What is the probability that it will snow at least one day while on their four-day ski trip?

The first step to calculate the chance of it snowing at least one day is to find the probability of it not snowing during their four-day trip. The resort says that there is a 65%, or 0.65 chance of it snowing each day. To find the probability of it not snowing, we will need to subtract 1 - 0.65, which equals 0.35. So the probability of it not snowing on any given day is 0.35.

Since they are going to be there for four days, we will need to multiply 0.35 times itself four times to represent the chance of it not snowing at all. The easiest way to do this is by using exponents. We will need to calculate 0.35^4, which is 0.015. The probability that it will not snow at all over their four-day trip is 0.015.

To calculate the probability that it will snow at least one day, we need to calculate the complement of this event. To do so, we will subtract 1 - 0.015, which equals 0.985. Tim and Jane know that there is 0.985, or 98.5% chance that it will snow at least one day during their ski trip. As Tim and Jane arrive at the ski resort, snow begins to fall. It is the most beautiful sight they've ever seen.

Lesson Summary

So, to review, independent events are events that do not affect the outcome of subsequent events. In an independent event, each situation is separate from previous events. Occasionally when calculating independent events, it is only important that the event occurs at least once. This is referred to as the 'At Least One' Rule. To calculate the probability of an event occurring at least once, it will be the complement of the probability of the event never occurring. When calculating this amount, you can use exponents to multiply the amount of the events that occurred.