If a dice is rolled and a coin is tossed then what is the probability of getting a head and a 4

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Probability is a branch of mathematics that deals with the happening of a random event. It is used in Maths to predict how likely events are to happen.

The probability of any event can only be between 0 and 1 and it can also be written in the form of a percentage.

The probability of event A is generally written as P(A).

Here P represents the possibility and A represents the event. It states how likely an event is about to happen. The probability of an event can exist only between 0 and 1 where 0 indicates that event is not going to happen i.e. Impossibility and 1 indicates that it is going to happen for sure i.e. Certainty

If we are not sure about the outcome of an event, we take help of the probabilities of certain outcomes—how likely they occur. For a proper understanding of probability we take an example as tossing a coin:

There will be two possible outcomes—heads or tails.

The probability of getting heads is half. You might already know that the probability is half/half or 50% as the event is an equally likely event and is complementary so the possibility of getting heads or tails is 50%.

Formula of Probability

Probability of an event, P(A) = Favorable outcomes / Total number of outcomes

Some Terms of Probability Theory

• Experiment: An operation or trial done to produce an outcome is called an experiment.
• Sample Space: An experiment together constitutes a sample space for all the possible outcomes. For example, the sample space of tossing a coin is head and tail.
• Favorable Outcome: An event that has produced the required result is called a favorable outcome.  For example, If we roll two dice at the same time then the possible or favorable outcomes of getting the sum of numbers on the two dice as 4 are (1,3), (2,2), and (3,1).
• Trial: A trial means doing a random experiment.
• Random Experiment: A random experiment is an experiment that has a well-defined set of outcomes. For example, when we toss a coin, we would get ahead or tail but we are not sure about the outcome that which one will appear.
• Event: An event is the outcome of a random experiment.
• Equally Likely Events: Equally likely events are rare events that have the same chances or probability of occurring. Here The outcome of one event is independent of the other. For instance, when we toss a coin, there are equal chances of getting a head or a tail.
• Exhaustive Events: An exhaustive event is when the set of all outcomes of an experiment is equal to the sample space.
• Mutually Exclusive Events: Events that cannot happen simultaneously are called mutually exclusive events. For example, the climate can be either cold or hot. We cannot experience the same weather again and again.
• Complementary Events: The Possibility of only two outcomes which is an event will occur or not. Like a person will eat or not eat the food, buying a bike or not buying a bike, etc. are examples of complementary events.

Some Probability Formulas

Addition rule: Union of two events, say A and B, then

P(A or B) = P(A) + P(B) – P(A∩B)

P(A ∪ B) = P(A) + P(B) – P(A∩B)

Complementary rule: If there are two possible events of an experiment so the probability of one event will be the Complement of another event. For example – if A and B are two possible events, then

P(B) = 1 – P(A) or P(A’) = 1 – P(A).

P(A) + P(A′) = 1.

Conditional rule: When the probability of an event is given and the second is required for which first is given, then

 P(B, given A) = P(A and B), P(A, given B). It can be vice versa

P(B∣A) = P(A∩B)/P(A)

Multiplication rule: Intersection of two other events i.e. events A and B need to occur simultaneously. Then 

P(A and B) = P(A)⋅P(B).

P(A∩B) = P(A)⋅P(B∣A)

Solution:

Use the binomial distribution. 

Lets suppose that the number of heads is r that represents the head times and in this case r = 4 

Assuming that the coin is unbiased, you have a probability of success ‘p’(where p is considered as success) is 1/2 and the probability of failure ‘q’ is (where q is considered as failure). 

The number of trials is represented by the ’n’ and here n = 12.

Now use the function for a binomial distribution:

P(R = r) = nCrprqn-r

P(R = 4) = (12C4)(1/2)4(1/2)12-4  

              = 495/4096

So the probability of flipping a coin 12 times and getting heads 4 times is 495/4096.

Similar Questions

Question 1: What are the chances of flipping 10 heads in a row?

Solution:

Probability of an event =  (number of favorable event) / (total number of event).

P(B) = (occurrence of Event B) / (total number of event).

Probability of getting one head = 1/2.

here Tossing a coin is an independent event, its not dependent on how many times it has been tossed.

Probability of getting 2 heads in a row = probability of getting head first time × probability of getting head second time.

Probability of getting 2 head in a row  = (1/2) × (1/2).

Therefore, the probability of getting 10 heads in a row = (1/2)10.

Question 2: What are the chances of flipping 12 heads in a row?

Solution:

Probability of an event = (number of favorable event) / (total number of event).

P(B) = (occurrence of Event B) / (total number of event).

Probability of getting one head = 1/2.

here Tossing a coin is an independent event, its not dependent on how many times it has been tossed.

Probability of getting 3 heads in a row = probability of getting head first time × probability of getting head second time x probability of getting head third time

Probability of getting 3 head in a row = (1/2) × (1/2) × (1/2)

Therefore, the probability of getting 20 heads in a row = (1/2)12

Question 3: If a coin is flipped 20 times, what is the probability of getting 6 heads?

Solution:

Use the binomial distribution.

Lets suppose that the number of heads is r that represents the head times and in this case r = 6

Assuming that the coin is unbiased, you have a probability of success ‘p’(where p is considered as success) is 1/2 and the probability of failure ‘q’ is (where q is considered as failure).

The number of trials is represented by the ’n’ and here n = 20.

Now use the function for a binomial distribution:

P(R = r) = nCr pr qn-r

P(R = 6) = (20C6)(1/2)6(1/2)12-6  

              = (20C6) (1/2)20

              = 38760/1048576

               = .0369644

So the probability of flipping a coin 20 times and getting heads 6 times is 0.0369644

Life is full of random events!

You need to get a "feel" for them to be a smart and successful person.

The toss of a coin, throwing dice and lottery draws are all examples of random events.

There can be:

Dependent Events where what happens depends on what happened before, such as taking cards from a deck makes less cards each time (learn more at Conditional Probability), or

Independent Events which we learn about here.

Independent Events are not affected by previous events.

This is an important idea!

A coin does not "know" it came up heads before.

And each toss of a coin is a perfect isolated thing.

The chance is simply ½ (or 0.5) just like ANY toss of the coin.

What it did in the past will not affect the current toss!

Some people think "it is overdue for a Tail", but really truly the next toss of the coin is totally independent of any previous tosses.

Saying "a Tail is due", or "just one more go, my luck is due to change" is called The Gambler's Fallacy

Of course your luck may change, because each toss of the coin has an equal chance.

Probability of Independent Events

"Probability" (or "Chance") is how likely something is to happen.

So how do we calculate probability?

Probability of an event happening = Number of ways it can happen Total number of outcomes

Example: what is the probability of getting a "Head" when tossing a coin?

Number of ways it can happen: 1 (Head)

Total number of outcomes: 2 (Head and Tail)

So the probability = 1 2 = 0.5

Example: what is the probability of getting a "4" or "6" when rolling a die?

Number of ways it can happen: 2 ("4" and "6")

Total number of outcomes: 6 ("1", "2", "3", "4", "5" and "6")

So the probability = 2 6 = 1 3 = 0.333...

Ways of Showing Probability

Probability goes from 0 (imposssible) to 1 (certain):

It is often shown as a decimal or fraction.

Example: the probability of getting a "Head" when tossing a coin:

• As a decimal: 0.5
• As a fraction: 1/2
• As a percentage: 50%
• Or sometimes like this: 1-in-2

We can calculate the chances of two or more independent events by multiplying the chances.

For each toss of a coin a Head has a probability of 0.5:

And so the chance of getting 3 Heads in a row is 0.125

So each toss of a coin has a ½ chance of being Heads, but lots of Heads in a row is unlikely.

Because we are asking two different questions:

Question 1: What is the probability of 7 heads in a row?

Answer: 12×12×12×12×12×12×12 = 0.0078125 (less than 1%)

Question 2: When we have just got 6 heads in a row, what is the probability that the next toss is also a head?

Answer: ½, as the previous tosses don't affect the next toss

You can have a play with the Quincunx to see how lots of independent effects can still have a pattern.

Notation

We use "P" to mean "Probability Of",

So, for Independent Events:

P(A and B) = P(A) × P(B)

Probability of A and B equals the probability of A times the probability of B

What are the chances you get Saturday between 4 and 6?

Day: there are two days on the weekend, so P(Saturday) = 0.5

Time: you want the 2 hours of "4 to 6", out of the 8 hours of 4 to midnight):

P("4 to 6") = 2/8 = 0.25

And:

Or a 12.5% Chance

(Note: we could ALSO have worked out that you wanted 2 hours out of a total possible 16 hours, which is 2/16 = 0.125. Both methods work here.)

Another Example

The chance of a flight not having a delay is 1 − 0.2 = 0.8, so these are all the possible outcomes:

When we add all the possibilities we get:

0.64 + 0.16 + 0.16 + 0.04 = 1.0

They all add to 1.0, which is a good way of checking our calculations.

Result: 0.64, or a 64% chance of no delays

One More Example

Imagine there are two groups:

• A member of each group gets randomly chosen for the winners circle,
• then one of those gets randomly chosen to get the big money prize:

What is your chance of winnning the big prize?

• there is a 1/5 chance of going to the winners circle
• and a 1/2 chance of winning the big prize

So you have a 1/5 chance followed by a 1/2 chance ... which makes a 1/10 chance overall:

15 × 12 = 15 × 2 = 110

Or we can calculate using decimals (1/5 is 0.2, and 1/2 is 0.5):

0.2 x 0.5 = 0.1

So your chance of winning the big money is 0.1 (which is the same as 1/10).

Coincidence!

Many "Coincidences" are, in fact, likely.

Do you say:

• "Wow, how strange !", or
• "That seems reasonable, with so many people here"

In fact there is a 70% chance that would happen ... so it is likely.

Why is the chance so high?

Because you are comparing everyone to everyone else (not just one to many).

And with 30 people that is 435 comparisons

(Read Shared Birthdays to find out more.)

Did you ever say something at exactly the same time as someone else?

Wow, how amazing!

But you were probably sharing an experience (movie, journey, whatever) and so your thoughts were similar.

And there are only so many ways of saying something ...

... so it is like the card game "Snap!" (also called Slaps or Slapjack) ...

... if you speak enough words together, they will eventually match up.

So, maybe not so amazing, just simple chance at work.

Can you think of other cases where a "coincidence" was simply a likely thing?

Conclusion

• Probability is: (Number of ways it can happen) / (Total number of outcomes)
• Dependent Events (such as removing marbles from a bag) are affected by previous events
• Independent events (such as a coin toss) are not affected by previous events
• We can calculate the probability of two or more Independent events by multiplying
• Not all coincidences are really unlikely (when you think about them).

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