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14.5 Probability identities
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14.7 Complementary events
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Two events are called mutually exclusive if they cannot occur at the same time. Whenever an outcome of an experiment is in the first event it cannot also be in the second event, and vice versa.
Another way of saying this is that the two event sets, \(A\) and \(B\), cannot have any elements in common, or \(P\left(A\cap B\right) = \varnothing\) (where \(\varnothing\) denotes the empty set). We have already seen the Venn diagrams of mutually exclusive events in the middle column of the Venn diagrams provided earlier.
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From this figure you can see that the intersection has no elements. You can also see that the probability of the union is the sum of the probabilities of the events.
\[P\left(A\cup B\right) = P\left(A\right) + P\left(B\right)\]This relationship is true for mutually exclusive events only.
We roll two dice and are interested in the following two events:
\(A:\) The sum of the dice equals \(\text{8}\)
\(B:\) At least one of the dice shows
Show that the events are mutually exclusive.
From the above figure we notice that there are no elements in common in A and B. Therefore the events are mutually exclusive.
State whether the following events are mutually exclusive or not.
A fridge contains orange juice, apple juice and grape juice. A cooldrink is chosen at random from the fridge. Event A: the cooldrink is orange juice. Event B: the cooldrink is apple juice.
We are choosing just one cooldrink from the fridge. This cooldrink cannot be both an orange juice and an apple juice. Therefore these two events are mutually exclusive.
A packet of cupcakes contains chocolate cupcakes, vanilla cupcakes and red velvet cupcakes. A cupcake is chosen at random from the packet. Event A: the cupcake is red velvet. Event B: the cupcake is vanilla.
We are choosing just one cupcake from the packet. This cupcake cannot be both a red velvet cupcake and a vanilla one. Therefore these two events are mutually exclusive.
A card is chosen at random from a deck of cards. Event A: the card is a red card. Event B: the card is a picture card.
We are choosing just one card from the deck. This card can be both a red card and a picture card. Therefore these two events are not mutually exclusive.
A cricket team plays a game. Event A: they win the game. Event B: they lose the game.
The cricket team can either win the game or lose the game. They cannot simultaneously win and lose the game. Therefore these two events are mutually exclusive.
Note that a tie game does not count as either a win or a loss. In a tie neither team can be said to have won the match.
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14.5 Probability identities
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