 Show Mutually Exclusive: can't happen at the same time. Examples:
What is not Mutually Exclusive:
Like here:
ProbabilityLet's look at the probabilities of Mutually Exclusive events. But first, a definition:
Probability of an event happening = Number of ways it can happen Total number of outcomes
Number of ways it can happen: 4 (there are 4 Kings) Total number of outcomes: 52 (there are 52 cards in total) So the probability = 4 52 = 1 13 Mutually ExclusiveWhen two events (call them "A" and "B") are Mutually Exclusive it is impossible for them to happen together: P(A and B) = 0 "The probability of A and B together equals 0 (impossible)"
A card cannot be a King AND a Queen at the same time!
But, for Mutually Exclusive events, the probability of A or B is the sum of the individual probabilities: P(A or B) = P(A) + P(B) "The probability of A or B equals the probability of A plus the probability of B"
In a Deck of 52 Cards:
When we combine those two Events:
Which is written like this: P(King or Queen) = (1/13) + (1/13) = 2/13 So, we have:
Special NotationInstead of "and" you will often see the symbol ∩ (which is the "Intersection" symbol used in Venn Diagrams) Instead of "or" you will often see the symbol ∪ (the "Union" symbol) So we can also write:
Example: Scoring GoalsIf the probability of:
Then:
Which is written: P(A ∩ B) = 0 P(A ∪ B) = 20% + 15% = 35% RememberingTo help you remember, think:
"Or has more ... than And" Also ∪ is like a cup which holds more than ∩ Not Mutually ExclusiveNow let's see what happens when events are not Mutually Exclusive. Example: Hearts and Kings
But Hearts or Kings is:
But that counts the King of Hearts twice! So we correct our answer, by subtracting the extra "and" part: 16 Cards = 13 Hearts + 4 Kings − the 1 extra King of Hearts Count them to make sure this works! As a formula this is: P(A or B) = P(A) + P(B) − P(A and B) "The probability of A or B equals the probability of A plus the probability of B Here is the same formula, but using ∪ and ∩: P(A ∪ B) = P(A) + P(B) − P(A ∩ B) A Final Example16 people study French, 21 study Spanish and there are 30 altogether. Work out the probabilities! This is definitely a case of not Mutually Exclusive (you can study French AND Spanish). Let's say b is how many study both languages:
And we get: And we know there are 30 people, so: (16−b) + b + (21−b) = 30 37 − b = 30 b = 7 And we can put in the correct numbers: So we know all this now:
Lastly, let's check with our formula: P(A or B) = P(A) + P(B) − P(A and B) Put the values in: 30/30 = 16/30 + 21/30 − 7/30 Yes, it works! Summary:Mutually Exclusive
Not Mutually Exclusive
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Despite the fact that (for good reason) this is not how the dealing is done, let us assume A gets the first $13$ cards, and B gets the next $13$, and so on. There are $\binom{52}{8}$ equally likely ways to choose the locations of the places where the $8$ cards we are interested in will go, and for each of these ways there are $8!$ ways to permute the cards. There are $13^4$ ways to choose a Queen location in each group of $13$. For each of these, there are $12^4$ ways to choose a King location in each group. The chosen Queen locations can be filled with Queens in $4!$ ways. For each of these, the chosen King locations can be filled with Kings in $4!$ ways. Thus the number of "favourables" is $13^412^44!4!$, and the required probability is $$\frac{13^412^4 4!4!}{\binom{52}{8}8!}.$$ Probability is a field of mathematics that studies the likelihood of a random event occurring. Since many events cannot be predicted with total certainty, we use probability to anticipate how probable they are to occur. Probability can range from 0 to 1, with 0 indicating an improbable event and 1 indicating a certain event. Probability has many applications. Risk assessment and modeling are examples of how probability theory is used in everyday life. Actuarial science is used by the insurance sector and markets to establish pricing and make trading decisions. Environmental control, entitlement analysis, and financial regulation all use probability methodologies. Probability also finds its applications in weather forecasting, agriculture, and politics. Formula for Probability
There are majorly three types of probability, they are theoretical probability, experimental probability, and axiomatic probability. Let’s learn about them in detail, Theoretical Probability It is predicated on the likelihood of something occurring. The rationale behind probability is the foundation of theoretical probability. For example, to calculate the theoretical probability of rolling a die and getting the number 3, we must first know the number of possible outcomes. We know that a die has six numbers (i.e. 1, 2, 3, 4, 5, 6), hence the number of possible outcomes is six as well. So, the probability of rolling a three on a dice is one in six, or 1/6 Experimental Probability Experimental probability, unlike theoretical probability, incorporates the number of trials, i.e. it is based on the results of an experiment. The experimental probability can be computed by dividing the total number of trials by the number of possible outcomes. For example, if a dice is rolled 40 times and the number three is recorded 10 times then, the experimental probability for heads is 10/40 or 1/4 Axiomatic Probability A set of principles or axioms are established in the axiomatic probability that applies to all types. Kolmogorov established these axioms, which are known as Kolmogorov’s three axioms. There are three main concepts in probability. They are sample space, events, and probability function. Let’s learn about them in detail, Sample Space (S) A sample space is the collection of all possible outcomes of an experiment. Tossing three dice produces a sample space of 216 potential outcomes, each of which may be recognized by an ordered set (a, b, c), where a, b, and c take one of the following values: 1, 2, 3, 4, 5, 6. Event (A) A well-defined subset of the sample space is referred to as an event. The event the sum of the faces shown on the two dice equals five has six outcomes: (1, 1, 3), (1, 3, 1), (3, 1, 1), (1, 2, 2), (2, 2, 1) and (2, 1, 1). Probability Function (P) The function that is used to assign a probability to an occurrence is known as the Probability Function (P). The probability function (P) determines the likelihood of an event (A) being drawn from the sample space (S). Solution:
Similar QuestionsQuestion 1: Find the probability of getting a red king. Solution:
Question 2: Find the probability of getting a red non-face card. Solution:
Question 3: Find the probability of getting a black card. Solution:
Question 4: Find the probability of getting a red ace or a spade. Solution:
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What is the probability of picking a card from a deck of 52 cards and getting a king or a queen?
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