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Solving Probability with Multiple Events

By Kathleen Knowles, 07 Oct 2020

What is the probability of two events happening? My friend asked me, “I need a clear day with no rain on either a Saturday or Sunday to plant my garden. I don’t care which. There is typically a 30% chance of rain on any given day during the summer months.”

Since the probability of two events both happening is the product of each, 0.30 times 0.30 equals 0.09. I explained, "The probability that it will rain both days is 9%. Therefore, the probability you will have at least one nice day is 91%.”

Note that since probabilities are fractions, multiplying them makes answers smaller.  Both events occurring is less likely than only one of them occurring. The answer satisfied my friend, but it was a little oversimplified.

Independent or Not

The correct statement should have been. “The probability of two independent events both happening is the product of each.” My knowledge of meteorology is a little lacking here, but I suspect the probability it rains on the second day is affected by whether or not it rained the first day.

For example, water laying on the ground after the first rain may be evaporated to add to humidity the second day. Or golfers who get rained out on Saturday may curse the rain gods who reply with more rain on Sunday.

Playing Cards

We can look at this situation in a different light using a basic deck of playing cards. From a deck of 52 face cards, what's the probability of drawing two cards that are of the same suit?

You'll first want to ask the question, did you put the first card back into the deck before drawing the second (with or without replacement)? If you did put the first card back into the deck, the second draw is independent of the first. The probability the second draw matches the first is 13/52 or 1/4—a 25% probability. (There are 13 cards of each suit in a deck.)

But if you are still holding the first card, there are now only 12 cards of that suit and only 51 cards in the deck.  So the probability is 12/51 which is about 0.235 or just slightly less than 25%.

Drawing the Same Cards Multiple Times

Here is a slightly more complicated question.  What is the probability of drawing hearts twice?  The probability that the first card is a heart is 1/4.  With replacement, the second card also has a 1/4 probability of being a heart, so the product is 1/4 x 1/4 = 1/16 or 0.0625.

Without replacement, (you are still holding the first card) the probability is 1/4 x 12/51 = 0.059.

Notice the wording of the two questions:

  • What is the probability both are the same?
  • What is the probability both are hearts?

A similar question—after you draw the first card (without replacement), what’s the probability the second card is a heart?  If the first is a heart, the second probability is 12/51.  If the first is not a heart, the second probability is 13/51.

Predicting Gender in Children

A friend told me she has 10 grandchildren, all of whom are girls. “What’s the probability of that?” she asked.  Did she mean the probability of all girls, or all the same?  I think the more logical question would be “all the same” because she would have been just as amazed had there been 10 boys.

What is the probability of 10 babies being all girls?

Around the world about 49% of babies at birth are girls and 51% are boys. This makes for very interesting demographic studies, and differences can be found based on characteristics of mothers such as age, race, health, and economic status.

Let's assume the birth rates are both 50%.  We will also assume that the gender selection of each child is independent, which may or may not be true, based on the technique of getting babies started.  The probability of 10 girls is 1/2 x 1/2 ten times or (0.5)10 = 1/1024 = 0.00098.

What is the probability of 10 babies being all the same gender?

Now it can be 10 girls or 10 boys.  When the first baby is born, he/she sets the standard for the rest.  The other 9 must match the first so there are 9 trials.  (Perhaps “trial” is not the right word in deference to those mothers having the babies.)  The probability is (0.5)9 = 1/512 = 0.00195.

Increased Restrictions or Possibilities

Consider the two characteristics “female” and “left-handed.” Roughly 50% of humans are female and about 10% are left-handed. (Assume gender and hand preference are independent.)  Here are four questions that can be asked.

What is the probability of a randomly selected human?

1) A left-handed female (ie. both left-handed and female)?  0.50 x 0.10 = 0.05.  The restrictions that both conditions must be met means the probability decreases.

2) Either left-handed or female?  Here it’s the possibilities that are increased rather than the restrictions.  The number of humans that meet either of the conditions is larger.  We will add probabilities.

However, if we consider all left-handed persons and then all females, the left-handed females get counted twice. So the probability is as follows:

  • Left-handed probability + female probability – probability of both (the result of Question 1 above).
  • 0.10 + 0.50 – 0.05 = 0.55.

3) Either left-handed or female, but not both?  We must subtract the probability of being both (Question 1) again.  0.55 – 0.05 = 0.50. This just happened to come out to be 50%, but it cannot be generalized.

4) Neither left-handed nor female?  This is the opposite of Question 2 above.  A person who is neither left-handed nor female, has to be a right-handed male.

Since 90% of humans are right-handed, the probability is 0.90 x 0.50 = 0.45.  Of all humans 45% are right-handed males, and 55% are not, either being left-handed or a female (or both).  Of course that adds up to 100% or 1 if expressed as a decimal.

Two Day’s Weather

Let’s return to the Saturday and Sunday weather question which is quite similar to the gender/preferred-hand question.  For simplicity's sake, let's assume the probability of rain is 30% for both days, and rain the second day is independent of the first. The probability of it not raining is 0.70 each day.

What's the probability that it rains both days?

(Rain, rain)  0.30 x 0.30 = 0.09

What's the probability that it rains one day

That means either Saturday or Sunday, but not both?  Either (rain, no rain) 0.30 x 0.70 = 0.21 or (no rain, rain) 0.7 x 0.30 = 0.21.

Since this is an increase of possibilities, (an “or” question) we add the two.  0.21 + 0.21 = 0.42. (There is no overlap where something was counted twice.)

Or we can use the method of Question 3 above: the sum of each, subtract the product twice.  0.30 + 0.30 – 2(0.09) = 0.42.

Same answer. (I love it when math is supposed to work and it does.)

What's the probability that there's one day with no rain?

All the possible results for the two days are (rain, rain), (rain, no rain), (no rain, rain), (no rain, no rain).  Of the four, the last three contain a no-rain day.  0.30 x 0.70 + 0.70 x 0.30 + 0.70 x 0.70 = 0.21 x 0.21 x 0.49 = 0.91.

Note that Questions 1 and 3 are opposites:

  • They include 100% of the possibilities
  • They're expressed as decimals
  • Their sum must add to 1

Often it is easier to calculate the probability of what you don’t want and subtract it from 1.  The opposite of at-least-one-good day is rain-both days.  1 – 0.09 = 0.91.

Conclusion

The probability that both events happen is the product of each if they're independent. If they're not, the probability of the second must be modified based on the results of the first. The probability that either one or the other happens is the sum of their probabilities, less the product of both if they overlap.  It may be easier to calculate 1 – the opposite of the desired probability.

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