 In this video, we're going to look at the and and all rules of probability and what the term mutually exclusive means. Firstly, do you know what mutually exclusive means? We use this term when we talk about two or more events or outcomes which cannot both occur at the same time. For example, the possible outcomes of a single coin toss you could get a head or a tail, but not both. Another example is the role of a standard dice. The outcomes are 1, 2, 3, 4, 5 and 6, but we can only have one, not more for the answer. We can also have events that are mutually non-exclusive. Two events are mutually non-exclusive if both the events have at least one common outcome between them. The two events cannot affect or prevent the occurrence of one another, so we can say they have something in common. For example, if you were to roll a dice with a desired outcome of an odd number and a number more than 4. The desired outcomes for events are 1, 3 and 5, whilst the desired outcomes for the other are 5 and 6. Because the outcome 5 is available for both, these events are called compatible and are mutually non-exclusive. Whether the events in a situation are exclusive or non-exclusive can affect the application of the further rules of probability. Let's look at these rules now. Firstly, the AND rule. The AND rule of probability is applied when we want to find the probability of two or more events happening. For example, the probability of rolling a 5 on a dice and flipping a coin and getting heads. The probability of rolling a 5 is 1 out of 6 because there are 6 possible outcomes on a dice. The probability of flipping heads on a coin is 1 out of 2 because there are 2 possible outcomes, heads and tails. When we compare these possible outcomes in a sample space, we can see that overall there are 12 combination outcomes. One of these outcomes is the outcome we wanted, a 5 and a head. Therefore, the probability of rolling a 5 and flipping heads is 1 out of 12. This is the exact same value as if we multiplied the two probabilities together. 1 over 6 multiplied 1 over 2 equals 1 over 12. Therefore, to find the probability of multiple events happening, we multiply their individual probabilities together. This will always result in a lower probability than the individual parts themselves, so remember to check that your combined probability is lower than the parts you started with. This fair six-sided spinner and this fair five-sided spinner are both spun. What's the probability of both spinners landing in an orange section? Pause and have a go. The first spinner has 2 out of 6 chances for orange, whilst the second has 1 out of 5. Multiplying these two probabilities gives an answer of 2 over 30, which simplifies to 1 over 15. Theoretically, therefore, one of every 15 spins should result in double orange. Our second rule is the all rule. This rule is based around the addition of probabilities, when more than one outcome will satisfy. This rule does not require multiple events to be occurring, as is impacted on by the exclusivity of the events. For mutually exclusive events, the probability that the first event or the second event will occur is the sum of the probability of each event. For example, if you roll a dice, what's the probability of rolling a 1 or a 6? Pause now if you want to have a go. The probability of a 1 is 1 over 6, and the probability of a 6 is also 1 over 6. There are therefore 2 out of 6 outcomes that are what we want, so the probability of a 1 or a 6 is 2 over 6. This can be simplified to 1 over 3 and is the same as the sum of the individual probabilities. There is a slight adjustment to the all rule when dealing with mutually non-exclusive events, such as the following question. If you would like to have a go, pause now and try it. What's the probability of getting a club or a jack from a shuffled pack of 52 cards? We can identify the number of both clubs and jacks in a pack fairly simply. There are 13 clubs, so this probability is 13 over 52. The number of jacks in a pack is 4, so this probability is 4 over 52. To find the total probability, if these were mutually exclusive, we would add them together, giving us 17 over 52 for our answer. However, these events are mutually non-exclusive, as this diagram here shows. There are only 16 cards, not 17. Because there is a shared outcome, the jack of clubs himself, we have to include an additional step, subtracting the probability of outcomes that satisfy both probabilities. The answer is therefore 16 over 52 or 4 over 13 in its simplified form.