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Odds For An Event Comments (44)
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Sometimes people express the likelihood of events in terms of odds rather than probabilities. You can solve any probability problem in terms of odds rather than probabilities.
Be sure that your numbers match the comparison. We can use odds to calculate how likely an event is to happen. You can use these two facts to compute the ratio of things happening and not happening.
For example, suppose the weather forecast states:. You can use this idea in many different situations. We know that the odds of it raining is 4 to 5.
What are the odds that it will as a percentage? Skip to main content. Search for:. Take a look at this dilemma: Telly and Carey were already hard at work when Ms.
Probability theory is an interesting area of statistics concerned with the odds or chances of an event happening in a trial, e. To work out odds, we also need to have an understanding of permutations and combinations.
The math isn't terribly complicated, so read on and you might be enlightened! What's covered in this guide:. This is determined by carrying out a series of trials.
So, for instance, a batch of products is tested and the number of faulty items is noted plus the number of acceptable items. Example: A sample of products is tested and 4 faulty items are found.
What is the probability of a product being faulty? This is a theoretical probability which can be worked out mathematically.
In this example, there is only 1 way a 6 can occur and there are 6 possible outcomes, i. Example 2: What is the probability of drawing a 4 from a pack of cards in one trial?
Public domain image via Pixabay. Once a probability has been worked out, it's possible to get an estimate of how many events will likely happen in future trials.
This is known as the expectation and is denoted by E. If the event is A and the probability of A occurring is P A , then for N trials, the expectation is:.
So in 60 trials, the expectation or number of expected 6's is:. Remember, the expectation is not what will actually happen, but what is likely to happen.
In 2 throws of a dice, the expectation of getting a 6 not two sixes is:. However, as we all know, it's quite possible to get 2 sixes in a row, even though the probability is only 1 in 36 see how this is worked out later.
As N becomes larger, the actual number of events which happen will get closer to the expectation. So for example when flipping a coin, if the coin isn't biased, the number of heads will be closely equal to the number of tails.
Events are independent when the occurrence of one event doesn't affect the probability of the other event. Two events are dependent if the occurrence of the first event affects the probability of occurrence of the second event.
Mutually exclusive events are events that cannot occur together. For instance in the throwing of a dice, a 5 and a 6 can't occur together.
Another example is picking coloured sweets out of a jar. Mutually non-exclusive event s are events that can occur together. For instance when a card is drawn from a pack and the event is a black card or an ace card.
If a black is drawn, this doesn't exclude it from being an ace. Similarly if an ace is drawn, this doesn't exclude it from being a black card.
Example 1: A sweet jar contains 20 red sweets, 8 green sweets and 10 blue sweets. If two sweets are pickets are picked out, what is the probability of picking a red or a blue sweet?
Example 2 : A dice is thrown and a card is drawn from a pack, what is the possibility of getting a 6 or an ace? There are 52 cards in a pack and four ways of getting an ace.
Also drawing an ace is an independent event to getting a 6 the earlier event doesn't influence it.
Remember in these type of problems, how the question is phrased is important. So the question was to determine the probability of one event occurring " or " the other event occurring and so the addition law of probability is used.
We effectively have to subtract the mutual events that are "double counted". You can think of the two probabilities as sets and we are removing the intersection of the sets and calculating the union of set A and set B.
Example 3: A coin is flipped twice. Calculate the probability of getting a head in either of the two trials.
Let H 1 be the event of a head in the first trial and H 2 be the event of a head in the second trial.
For more information on mutually non-exclusive events, see this article: Taylor, Courtney. Example: A dice is thrown and a card drawn from a pack, what is the probability of getting a 5 and a spade card?
There are 52 cards in the pack and 4 suits or groups of cards, aces, spades, clubs and diamonds. Each suit has 13 cards, so there are 13 ways of getting a spade.
Again it's important to note that the word " and " was used in the question, so the multiplication law was used. Engineering Mathematics by K. Stroud is an excellent math textbook for both engineering students and anyone with a general interest in mathematics.
The material has been written for part 1 of BSc. Engineering Degrees and Higher National Diploma courses. A wide range of topics are covered including matrices, vectors, complex numbers, calculus, calculus applications, differential equations and series.
The text is written in the style of a personal tutor, guiding the reader through the content, posing questions and encouraging them to provide the answer.
Personally, I've found it really easy to follow. It also covers a more in-depth treatment of probability theory than what has been covered in this article plus a section on statistics.
This book basically makes learning mathematics fun! It follows from rule 2 that the probability of an event not occurring is 1 - the probability of it occurring:.
To solve more difficult problems and derive an expression for the probability of a general binomial distribution, we need to understand the concept of permutations and combinations.
I won't go into the mathematics of the derivation, but basically the expression is derived from the equation for working out combinations.
A permutation is a way of arranging a number of objects. So, for instance, if you have the letters A, B, and C then all the possible permutations are:.
If you have n objects, there are n factorial number of ways of arranging them, written as n! The reason for this is because for the first position, there are n choices, and for each of these choices, there are n-1 choices for the second place because 1 choice was used up for the first place , and for each of the choices in the first two places, n-3 choices for the third place and so on.
Example: 2 letters are chosen from the set of letters A, B, C, D. How many ways can the 2 letters be arranged? Public domain via Pixabay.
A combination is a way of selecting objects from a set without regard to the order of the objects. So again if we have the letters A, B and C and select 3 letters from this set, there is only 1 way of doing this, i.
In general, if you have n objects in a set and make selections r at a time, the total possible number of selections is:.
Example: 2 letters are chosen from the set ABCD. How many combinations are possible? We would all like to win the lottery, but the chances of winning are only slightly greater than 0.