When calculating odds, you're trying to figure out the probability of an event occurring for a given number of trials. Probability is the probability that one or more events will occur divided by the number of possible outcomes. Calculating the probability of occurrence of several events is done by dividing the problem into several probabilities and multiplying them by each other.
Step
Method 1 of 3: Finding the Chance of One Random Event
Step 1. Select events with mutually exclusive outcomes
Odds can only be calculated when the event (for which the odds are calculated) occurs or does not occur. Events and their opposites cannot occur at the same time. Rolling the number 5 on the dice, the horse that wins the race, is an example of a mutually exclusive event. Either you roll the number 5, or you don't; either your horse wins the race, or not.
Example:
It is impossible to calculate the probability of an event: "The numbers 5 and 6 will appear on one roll of the dice."
Step 2. Determine all the possible events and outcomes that could occur
Say you are trying to find the probability of getting the numbers 3 and 6 on the dice. "Rolling the number 3" is an event, and since a 6-sided die can turn up any of the numbers 1-6, the number of outcomes is 6. So, in this case we know that there are 6 possible outcomes and 1 event whose odds we want count. Here are 2 examples to help you:
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Example 1: What is the probability of getting a day that falls on the weekend when choosing a day at random?
"Selecting a day that falls on the weekend" is an event, and the number of results is the total day of the week, which is 7.
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Example 2: The jar contains 4 blue marbles, 5 red marbles, and 11 white marbles. If one marble is drawn from the jar at random, what is the probability that a red marble is drawn?
"Selecting the red marbles" is our event, and the number of results is the total number of marbles in the jar, which is 20.
Step 3. Divide the number of events by the total number of results
This calculation will show the probability that one event will occur. In the case of rolling a 3 on a 6-sided die, the number of events is 1 (there is only one 3 in the die), and the number of outcomes is 6. You can also express this relationship as 1 6, 1/6, 0, 166, or 16, 6%. Check out some other examples below:
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Example 1: What is the probability of getting a day that falls on the weekend when choosing a day at random?
The number of events is 2 (since the weekend consists of 2 days), and the number of outcomes is 7. The probability is 2 7 = 2/7. You can also express it as 0.285 or 28.5%.
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Example 2: The jar contains 4 blue marbles, 5 red marbles, and 11 white marbles. If one marble is drawn from the jar at random, what is the probability that a red marble is drawn?
The number of events is 5 (since there are 5 red marbles), and the sum of the outcomes is 20. Thus, the probability is 5 20 = 1/4. You can also express it as 0, 25 or 25%.
Step 4. Add up all probability events to make sure they equal 1
The probability of occurrence of all events must reach 1 aka 100%. If the odds don't reach 100%, it's likely that you made a mistake because there was a missed opportunity event. Double-check your calculations for errors.
For example, your probability of getting a 3 when you roll a 6-sided die is 1/6. However, the odds of rolling the other five numbers on the dice are also 1/6. 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 6/6, which is equal to 100%
Notes:
For example, if you have forgotten to include the odds of the number 4 on the dice, the total odds are only 5/6 or 83%, indicating an error.
Step 5. Give 0 for the impossible chance
This means that the event will never come true, and appears every time you handle an impending event. While calculating 0 odds is rare, it's not impossible either.
For example, if you calculate the probability that the Easter holiday falls on a Monday in 2020, the probability is 0 because Easter is always celebrated on a Sunday
Method 2 of 3: Calculating the Probability of Multiple Random Events
Step 1. Handle each opportunity separately to calculate independent events
Once you know what the odds of each event are, calculate them separately. Say you want to know the probability of rolling the number 5 twice in a row on a 6-sided die. You know that the probability of rolling the number 5 once is, and the probability of rolling the number 5 again is also. The first result does not interfere with the second result.
Notes:
The probability of getting the number 5 is called independent event because what happens the first time doesn't affect what happens the second time.
Step 2. Consider the impact of previous events when calculating dependent events
If the occurrence of one event changes the probability of the second event, you are calculating the probability dependent event. For example, if you have 2 cards from a deck of 52 cards, when you select the first card, this affects the odds of the cards that can be drawn from the deck. To calculate the probability of a second card from two dependent events, subtract the number of possible outcomes by 1 when calculating the probability of the second event.
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Example 1: Consider an event: Two cards are drawn at random from the card deck. What is the probability that both are cards of spades?
The odds of the first card having the spade symbol are 13/52, or 1/4. (There are 13 cards of spades in a complete card deck).
Now, the probability of the second card having the spade symbol is 12/51 because 1 of the spades has already been drawn. Thus, the first event affects the second event. If you draw a 3 of spades and don't put it back in the deck, it means that the spade card and the deck's total are reduced by 1 (51 instead of 52)
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Example 2: The jar contains 4 blue marbles, 5 red marbles, and 11 white marbles. If 3 marbles are drawn at random from the jar, what is the probability that a red marble, a blue second marble, and a white third marble are drawn?
The probability of drawing a red marble the first time is 5/20, or 1/4. The probability of drawing a blue color for the second marble is 4/19 because the total number of marbles in the jar is reduced by one, but the number of blue marbles has not decreased. Finally, the probability that the third marble is white is 11/18 because you have already selected 2 marbles
Step 3. Multiply the probabilities of each separate event from each other
Whether you are working on independent or dependent events, and the number of outcomes involved is 2, 3, or even 10, you can calculate the total probability by multiplying these separate events. The result is the probability of several events occurring one after another. So, for this scenario, what is the probability that you will roll 5 in a row on a six-sided die? The probability that one roll of the number 5 occurs is 1/6. Thus, you calculate 1/6 x 1/6 = 1/36. You can also present it as a decimal number of 0.027 or a percentage of 2.7%.
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Example 1: Two cards are drawn from the deck at random. What is the probability that both cards have the spade symbol?
The probability of the first event occurring is 13/52. The probability of the second event occurring is 12/51. The probability of both is 13/52 x 12/51 = 12/204 = 1/17. You can present it as 0.058 or 5.8%.
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Example 2: A jar containing 4 blue marbles, 5 red marbles, and 11 white marbles. If three marbles are drawn from the jar at random, what is the probability that the first marble is red, the second is blue, and the third is white?
The probability of the first event is 5/20. The probability of the second event is 4/19. Lastly, the odds of a third event are 11/18. The total odds are 5/20 x 4/19 x 11/18 = 44/1368 = 0.032. You can also express it as 3.2%.
Method 3 of 3: Turning Opportunities Into Probability
Step 1. Present the probability as a ratio with a positive result as the numerator
For example, let's look again at the example of a jar filled with colored marbles. Say you want to know the probability that you will draw a white marble (of which there are 11), from the total number of marbles in the jar (of which there are 20). The probability of an event occurring is the ratio of the probability of an event will happen to the probability will not happen. Since there are 11 white marbles and 9 non-white marbles, the odds are written in the ratio 11:9.
- The number 11 represents the probability of drawing a white marble and the number 9 represents the probability of drawing a marble of another color.
- So, your chances of pulling white marbles are quite high.
Step 2. Add up the numbers to turn the odds into probabilities
Changing the odds is quite simple. First, break the probability into 2 separate events: the probability of drawing a white marble (11) and the probability of drawing another colored marble (9). Add the numbers together to calculate the total number of results. Write it down as a probability, with the new total number calculated as the denominator.
The number of outcomes from the event that you pick a white marble is 11; the number of results you draw other colors is 9. So the total number of results is 11 + 9, or 20
Step 3. Find the probability as if you were calculating the probability of a single event
You have seen that there are a total of 20 possibilities, and 11 of them are to draw white marbles. So, the probability of drawing a white marble can now be worked out like dealing with the probability of any other event. Divide 11 (number of positive outcomes) by 20 (total number of events) to get the probability.
So, in our example, the probability of drawing a white marble is 11/20. Divide the fraction: 11 20 = 0.55 or 55%
Tips
- Mathematicians usually use the term "relative frequency" to refer to the probability that an event will occur. The word “relative” is used because no outcome is 100% guaranteed. For example, if you flick a coin 100 times, possible You won't exactly get 50 sides of numbers and 50 sides of logos. Relative odds also take this into account.
- The probability of an event cannot be a negative number. If you get a negative number, double-check your calculations.
- The most common ways of presenting odds are with fractions, decimal numbers, percentages, or a 1–10 scale.
- You need to know that in sports betting, odds are expressed as “odds against” (odds against), which means that the odds of the event occurring are written first, and the odds of the event not occurring are written later. While it can be confusing at times, you need to know it if you want to try your luck at sporting events.
