How to find the probability of two events that might or might not intersect. When to add and subtract probabilities.
- Sometimes there are multiple outcomes that would lead us to the same conclusion. For example, suppose we flip two coins, you win if one or both of the coins turns up heads, what are your odds of winning? Well, let's look at all the possible outcomes. We can get heads on flip one, tails flip two. Heads flip one, heads flip two. Tails flip one, tails flip two. And finally, tails flip one, heads flip two.
We can see here that two different events will win this contest for you. Event one, heads of flip one, two out of the four scenarios provide that result. Event two, heads on flip two, same here, two out of the four scenarios provide that result, but we also see that there is an overlap here. We wanna be sure not to double-count that outcome. So to calculate the probability of getting heads on at least one of the two coin flips we add the probability of event one plus the probability of even two, but we subtract the overlap, which is when both event one and event two occur.
We can see that our probability of getting heads on at least one of two coin flips is 75%. This is what is called the addition rule. Let's up the difficulty level just a tiny bit. Let's do this with a pair of six-sided dice. You win if you roll either a six with roll one or roll two. Here are all the 36 possible outcomes. There are six outcomes where we roll six on the first die.
There are six outcomes where we roll six on the second die, but one of those outcomes overlaps. So to calculate the total probability we add 6/36ths plus 6/36ths and subtract the overlap, 1/36ths, thus the probability is 11/36ths or 30.56%. Sometimes there are no overlapping scenarios, for example, what are the odds that the sum of two rolled dice will sum to either seven or 11? Here are the scenarios where the dice add up to seven, here are the two scenarios where the dice add up to 11, notice there is no overlap.
So here there is no need to subtract anything, just add the odds of rolling a seven, 6/36ths, and add the odds of rolling 11, 2/36ths. Here the probability is 8/36ths or 22.2%. We can also flip this around. What are the odds of not rolling a seven or 11? Again, here are the eight scenarios where the sum of the two dice would be seven or 11.
We can count all the other scenarios, but since we know eight of 36 scenarios sum to seven or 11, we know that 28 of 36 scenarios do not sum up to seven or 11. The probability is 77.8%. These were all relatively simple scenarios, but hopefully as you go forward you'll remember you can add probabilities and you can also subtract probabilities.
Professor Eddie Davila covers statistics basics, like calculating averages, medians, modes, and standard deviations. He shows how to use probability and distribution curves to inform decisions, and how to detect false positives and misleading data. Each concept is covered in simple language, with detailed examples that show how statistics are used in real-world scenarios from the worlds of business, sports, education, entertainment, and more. These techniques will help you understand your data, prove theories, and save time, money, and other valuable resources—all by understanding the numbers.
- Calculate mean and median for specific data sets.
- Explain how the mode is used to assess a data set.
- Identify situations in which standard deviation can be used to investigate individual data points.
- Use mean and standard deviation to find the Z-score for a data point.
- List the three different categories of probability.
- Analyze data to determine if two events are dependent or independent.
- Predict possible outcomes for a situation using basic permutation calculations.
- Give examples of binomial random variables.