Question Video: Deciding If Events Are Mutually Exclusive | Nagwa Question Video: Deciding If Events Are Mutually Exclusive | Nagwa

# Question Video: Deciding If Events Are Mutually Exclusive Mathematics • Third Year of Preparatory School

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Amelia has a deck of 52 cards. She randomly selects one card and considers the following events: Event 𝐴: picking a card that is a heart. Event 𝐵: picking a card that is black. Event 𝐶: picking a card that is not a spade. Are events 𝐴 and 𝐵 mutually exclusive? Are events 𝐴 and 𝐶 mutually exclusive? Are events 𝐵 and 𝐶 mutually exclusive?

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### Video Transcript

Amelia has a deck of 52 cards. She randomly selects one card and considers the following events: event 𝐴, picking a card that is a heart; event 𝐵, picking a card that is black; and event 𝐶, picking a card that is not a spade. Are events 𝐴 and 𝐵 mutually exclusive? Are events 𝐴 and 𝐶 mutually exclusive? Are events 𝐵 and 𝐶 mutually exclusive?

In all three parts of this question, we need to determine whether two events are mutually exclusive. We recall that two events 𝑥 and 𝑦 are mutually exclusive if they cannot both occur at the same time, in other words, the probability of 𝑥 intersection 𝑦 is equal to zero.

In this question, we are told that Amelia has a regular deck of 52 cards. We know that these are split into four suits, diamonds, hearts, clubs, and spades, where the first two suits are red cards and the second two are black. Each of the suits has 13 cards: an ace, the numbers two through 10, a jack, a queen, and a king.

There are three events that Amelia needs to consider: firstly, event 𝐴, picking a card that is a heart. This would involve picking any of the 13 cards in the second row. Event 𝐵 involves picking a card that is black. This involves picking a card that is either a club or a spade, any of the 26 cards in the bottom two rows. Finally, we have event 𝐶, which is picking a card that is not a spade. This could be either a diamond, heart, or club, any of the 39 cards in the top three rows.

To determine whether events 𝐴 and 𝐵 are mutually exclusive, we need to find whether there is an outcome that occurs in both events. Is it possible to pick a card that is a heart and pick a card that is black? We know that all hearts are red, so there are no cards which are both black and a heart. And we can therefore conclude that when Amelia is selecting one card, both events cannot happen. And the events are therefore mutually exclusive.

Next, we need to consider whether events 𝐴 and 𝐶 are mutually exclusive. This time, we have the events of picking a card that is a heart and picking a card that is not a spade. The key point here is that all hearts are not spades. This means that choosing any heart will satisfy both events. And since both events can occur at the same time, we can conclude that they are not mutually exclusive. There is an overlap between picking a card that is a heart and picking a card that is not a spade.

Finally, we need to consider whether events 𝐵 and 𝐶 are mutually exclusive. This time, we have the events of picking a card that is black and picking a card that is not a spade. This time, the key fact is that all clubs are black. And they are also not a spade. This means that choosing any of the clubs satisfies both events. And we can therefore conclude that events 𝐵 and 𝐶 are not mutually exclusive. The events 𝐴 and 𝐵 are mutually exclusive, whereas events 𝐴 and 𝐶 and 𝐵 and 𝐶 are not mutually exclusive.

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