Probability of compound events

The probability of compound events combines at least two simple events. The probability that a coin will show head when you toss only one coin is a simple event.






However, if you toss two coins, the probability of getting 2 heads is a compound events because once again it combines two simple events

Suppose you say to a friend, " I will give you 10 dollars if both coins land on head."

Let's see what happens when your friend toss two coins:

If heads = H and tails = T, the different outcomes are HH, HT, TH, or TT.

As you can see, out of 4 possibilities, only 1 will give you HH.

Therefore, the probability of getting 2 heads is
1 / 4


Your friend has 25% chance of getting 10 dollars since one-fourth = 25%.

The example above is a good example of independent events. What are independent events?

When the outcome of one event does not affect the outcome of another event, the two events are said to be independent.

In our example above, when you toss two coins, neither coin has the power to influence the other coin.

This compound events is independent then. When two events are independent, you can use the following formula

probability(A and B) = probability(A) × probability(B)

Let's use this formula to find the probability of getting 2 heads when two coins are tossed

probability(H and H) = probability(H) × probability(H)

Coin #1:

Probability of getting head =
1 / 2


Coin #2:

Probability of getting another head =
1 / 2


probability(H and H) = probability(H) × probability(H)

probability(H and H) =
1 / 2
×
1 / 2


probability(H and H) =
1 × 1 / 2 × 2


probability(H and H) =
1 / 4




Sometimes, compound events can be dependent. What are dependents events?

When the outcome of one event has the power to affect the outcome of another event, the two events are said to be dependent.

When two events are dependent, you can use the following formula

probability(A and B) = probability(A) × probability(B given A)

Soppose, a bag has 4 red balls and 6 blue balls. What is the probability of choosing 2 blue balls at random?

These events are dependent since after you choose one blue ball, it changes the number of blue balls and the number of balls all together

Blue ball #1:

Probability of getting a blue ball =
6 / 10


Blue ball #2: Now there are 5 blue balls and 9 balls all together

Probability of getting another blue ball =
5 / 9


Let Blue = B

probability(B and B) = probability(B) × probability(B)

probability(B and B) =
6 / 10
×
5 / 9


Probability of getting B and B =
6 × 5 / 10 × 9


Probability of getting B and B =
30 / 90


Probability of getting B and B =
1 / 3


You have 33.33% chance of doing this since 1/3 is equal to 33.33%

Lastly, sometimes, as opposed to having two events happening at the same time, you may need to choose between two events.

When two events cannot both occur, they are called mutually exclusive events.

To find the probability of compound events when the events are mutually exclusive, use the formula:

probability (A or B) = probability (A) + probability (B)

Suppose you and your brother both throw a die. Whoever get a 4 wins!

These are mutually exclusive events because you cannot both win this game.

Let Y = you win and B = your brother win probability (Y or B) = probability (Y) + probability (B)

You:

Probability you win =
1 / 6


Your brother:

Probability your brother wins =
1 / 6


probability(Y or B) = probability(Y) + probability(B)

probability(Y or B) =
1 / 6
+
1 / 6


probability(Y or B) =
1 + 1 / 6


probability(Y or B) =
2 / 6








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