How to Understand Probability

Step-by-Step Guide

1. 1

   Step 1: Thin about the definition of probability.

   Probability is the likelihood of a random event happening.It is usually expressed as a ratio.
   
   Since probability is expressed as a ratio, or fraction, you can think of it as being the odds that something will happen, on a scale of 0 to 1, with 0 being no chance, and 1 being certain (that is, the event will happen 1 out of 1 times).Probability describes random events.
   
   A random event is an event that cannot be predicted,for example, pulling a particular card out of a deck, or getting struck by lightning.
2. 2

   Step 2: Understand the formula for determining probability.

   The probability of something happening is defined by the ratio probability=numberoffavorableoutcomesnumberofpossibleoutcomes{\displaystyle probability={\frac {number\;of\;favorable\;outcomes}{number\;of\;possible\;outcomes}}}, where a favorable outcome is the event you are seeking to happen., To do this, complete the probability ratio by determining how many favorable outcomes you can have, and how many possible outcomes you can have.
   
   Before you can understand more complex probability theory, you must understand how to figure out the probability of a single, random event happening, and understand what that probability means.
   
   For example, if you have a jar with 10 red marbles and 5 blue marbles, you might want to know what the possibility of randomly pulling out a blue marble is.
   
   Since you have 5 blue marbles, the number of favorable outcomes is
   5.
   
   Since you have 15 marbles total in your jar, the number of possible outcomes is
   15.
   
   Your probability ratio will look like this:probability=numberoffavorableoutcomesnumberofpossibleoutcomes{\displaystyle probability={\frac {number\;of\;favorable\;outcomes}{number\;of\;possible\;outcomes}}}probability=515{\displaystyle probability={\frac {5}{15}}} Simplified, probability=13{\displaystyle probability={\frac {1}{3}}}.
   
   So, the probability of randomly pulling out a blue marble is 1 out of
   3. , Independent events are ones in which the outcome of one event does not affect the probability of the other event happening.For example, if you are using two dice, you might want to know what the probability is that you will roll a double
   3.
   
   The chance that you will throw a 3 with one die does not affect the chance that you will throw a 3 with the second die, thus the events are independent. , To do this, set up the ratio probability=numberoffavorableoutcomesnumberofpossibleoutcomes{\displaystyle probability={\frac {number\;of\;favorable\;outcomes}{number\;of\;possible\;outcomes}}}, where a favorable outcome is the event you are seeking to happen.
   
   For example, if the first event is throwing a 3 with one die, the number of favorable outcomes is 1, since there is only one 3 on a die.
   
   The number of possible outcomes is 6, since a die has six sides.
   
   So, your ratio will look like this: probability=16{\displaystyle probability={\frac {1}{6}}}. , To do this, set up the ratio, just like you did for the first event.
   
   For example, if the second event is also throwing a 3 with one die, the probability is the same as the first event: probability=16{\displaystyle probability={\frac {1}{6}}}.
   
   The probability of the first and second event might not be the same.
   
   For example, if you and a classmate own the same outfit, you might want to know the probability that she and you will wear the same outfit to school on the same day.
   
   If you have five outfits, the odds of you wearing the outfit is 15{\displaystyle {\frac {1}{5}}}, but if your classmate has ten outfits, the odds of her wearing the outfit is 110{\displaystyle {\frac {1}{10}}}. , This will give you the probability of both events happening.For a refresher on how to multiply fractions, read Multiply Fractions.
   
   For example, if the probability of throwing a 3 with one die is 16{\displaystyle {\frac {1}{6}}}, and the probability of throwing a 3 with a second die is also 16{\displaystyle {\frac {1}{6}}}, to find the probability of both events happening, you would calculate 16×16=136{\displaystyle {\frac {1}{6}}\times {\frac {1}{6}}={\frac {1}{36}}}.
   
   So, the probability of throwing double threes is 1 out of
   36. , A conditional event, also called a dependent event, is an event that may be affected by the event(s) that come before.For example, if you are drawing from a standard deck of cards, you might want to know what the probability is of drawing a heart on the first and second draws.
   
   Drawing a heart the first time affects the probability of it happening again, because once you draw one heart, there are fewer hearts in the deck, and fewer cards in the deck. , To do this, set up the ratio probability=numberoffavorableoutcomesnumberofpossibleoutcomes{\displaystyle probability={\frac {number\;of\;favorable\;outcomes}{number\;of\;possible\;outcomes}}}, where a favorable outcome is the event you are seeking to happen.
   
   For example, if the first event is drawing a heart from a deck of cards, the number of favorable outcomes is 13, since there are 13 hearts in a deck.
   
   The number of possible outcomes is 52, since a deck has 52 cards total.
   
   So, your ratio will look like this: probability=1352{\displaystyle probability={\frac {13}{52}}}.
   
   Simplified, the probability is 14{\displaystyle {\frac {1}{4}}}. , For example, if you pulled a heart on your first draw, now there are only 12 hearts in the deck, and there are only 51 cards total.
   
   So, the probability of drawing a heart on your second draw is 1251{\displaystyle {\frac {12}{51}}}.
   
   Simplified, the probability is 417{\displaystyle {\frac {4}{17}}}. , This will give you the probability of both events happening.For a refresher on how to multiply fractions, read Multiply Fractions.
   
   For example, if the probability of pulling a heart on your first draw is 14{\displaystyle {\frac {1}{4}}}, and the probability of pulling a heart on your second draw, given that you pulled a heart on your first draw, is 417{\displaystyle {\frac {4}{17}}}, to find the probability of both events happening, you would calculate:14×417=468{\displaystyle {\frac {1}{4}}\times {\frac {4}{17}}={\frac {4}{68}}}468=117{\displaystyle {\frac {4}{68}}={\frac {1}{17}}}So, the probability of pulling hearts on your first and second draw is 1 out of
   17. , Mutually exclusive events are events that cannot happen at the same time.Mutually exclusive events will be marked by the conjunction or. (Events that are not mutually exclusive will use the conjunction and.)For example, if you are rolling one die, you might want to know the probability of rolling a 3 or a
   4.
   
   You cannot roll a 3 and a 4 with one die, so the events are mutually exclusive. , To do this, set up the ratio probability=numberoffavorableoutcomesnumberofpossibleoutcomes{\displaystyle probability={\frac {number\;of\;favorable\;outcomes}{number\;of\;possible\;outcomes}}}, where a favorable outcome is the event you are seeking to happen.
   
   For example, if the first event is throwing a 3 with one die, the number of favorable outcomes is 1, since there is only one 3 on a die.
   
   The number of possible outcomes is 6, since a die has six sides.
   
   So, your ratio will look like this: probability=16{\displaystyle probability={\frac {1}{6}}}. , To do this, set up the ratio, just like you did for the first event.
   
   For example, if the second event is throwing a 4 with one die, the probability is the same as the first event: probability=16{\displaystyle probability={\frac {1}{6}}}.
   
   The probability of the first and second event might not be the same.
   
   For example, you might want to know the probability of the next random song in a 32-song playlist being hip hop or folk.
   
   If there are 12 hip hop songs in the playlist, and 6 folk songs, the probability of the next song being hip hop is 1232{\displaystyle {\frac {12}{32}}}, and the probability of it being folks is 632{\displaystyle {\frac {6}{32}}}. , This will give you the probability of either event happening.
   
   For a refresher on how to add fractions, read Add Fractions.
   
   For example, if the probability of throwing a 3 with one die is 16{\displaystyle {\frac {1}{6}}}, and the probability of throwing a 4 with one die is also 16{\displaystyle {\frac {1}{6}}}, to find the probability of both events happening, you would calculate:16+16=26{\displaystyle {\frac {1}{6}}+{\frac {1}{6}}={\frac {2}{6}}}26=13{\displaystyle {\frac {2}{6}}={\frac {1}{3}}}So, the probability of throwing a 3 or a 4 is 1 out of
   3.
3. 3

   Step 3: Determine the probability of a single event happening.

4. 4

   Step 4: Determine whether the two events are independent.

5. 5

   Step 5: Determine the probability of the first event.

6. 6

   Step 6: Determine the probability of the second event.

7. 7

   Step 7: Multiply the probabilities of the individual events.

8. 8

   Step 8: Determine whether the two events are conditional.

9. 9

   Step 9: Determine the probability of the first event happening.

10. 10

    Step 10: Determine the probability of the second event happening

11. 11

    Step 11: given that the first event already happened.To do this

12. 12

    Step 12: you will need to examine how the first event happening will affect the number of favorable and possible outcomes of the second event.

13. 13

    Step 13: Multiply the probabilities of the individual events.

14. 14

    Step 14: Determine whether the two events are mutually exclusive.

15. 15

    Step 15: Determine the probability of the first event.

16. 16

    Step 16: Determine the probability of the second event.

17. 17

    Step 17: Add the probabilities of the individual events.

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