A fair coin is flipped 11 times find the probability that more than 5 of the flips turn up tails

The probability of some event happening is a mathematical (numerical) representation of how likely it is to happen, where a probability of 1 means that an event will always happen, while a probability of 0 means that it will never happen. Classical probability problems often need to you find how often one outcome occurs versus another, and how one event happening affects the probability of future events happening. When you look at all the things that may occur, the formula (just as our coin flip probability formula) states that

probability = (no. of successful results) / (no. of all possible results).

Take a die roll as an example. If you have a standard, 6-face die, then there are six possible outcomes, namely the numbers from 1 to 6. If it is a fair die, then the likelihood of each of these results is the same, i.e., 1 in 6 or 1 / 6. Therefore, the probability of obtaining 6 when you roll the die is 1 / 6. The probability is the same for 3. Or 2. You get the drill. If you don't believe me, take a dice and roll it a few times and note the results. Remember that the more times you repeat an experiment, the more trustworthy the results. So go on, roll it, say, a thousand times. We'll be waiting here until you get back to tell us we've been right all along. Go to the dice probability calculator if you want a shortcut.

But what if you repeat an experiment a hundred times and want to find the odds that you'll obtain a fixed result at least 20 times?

Let's look at another example. Say that you're a teenager straight out of middle school and decide that you want to meet the love of your life this year. More specifically, you want to ask ten girls out and go on a date with only four of them. One of those has got to be the one, right? The first thing you have to do in this situation is look in the mirror and rate how likely a girl is to agree to go out with you when you start talking to her. If you have problems with assessing your looks fairly, go downstairs and let your grandma tell you what a handsome, young gentleman you are. So a solid 9 / 10 then.

As you only want to go on four dates, that means you only want four of your romance attempts to succeed. This has an outcome of 9 / 10. This means that you want the other six girls to reject you, which, based on your good looks, has only a 1 / 10 change of happening (The sum of all events happening is always equal to 1, so we get this number by subtracting 9 / 10 from 1). If you multiply the probability of each event by itself the number of times you want it to occur, you get the chance that your scenario will come true. In this case, your odds are 210 * (9 / 10)4 * (1 / 10)6 = 0.000137781, where the 210 comes from the number of possible fours of girls among the ten that would agree. Not very likely to happen, is it? Maybe you should try being less beautiful!

We build a mathematical model of the experiment. Write H for head and T for tail. Record the results of the tosses as a string of length $10$, made up of the letters H and/or T. So for example the string HHHTTHHTHT means that we got a head, then a head, then a head, then a tail, and so on.

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• Video Transcript
• What is the probability of flipping a coin 3 times and getting tails?
• What is the probability of getting no more than 3 heads when you flip a fair coin 5 times?
• What is the probability of getting 3 heads or less in 10 tosses of a fair coin?
• What is the probability that when you flip a coin 3 times you will get 3 heads?

There are $2^{10}$ such strings of length $10$. This is because we have $2$ choices for the first letter, and for every such choice we have $2$ choices for the second letter, and for every choice of the first two letters, we have $2$ choices for the third letter, and so on.

Because we assume that the coin is fair, and that the result we get on say the first $6$ tosses does not affect the probability of getting a head on the $7$-th toss, each of these $2^{10}$ ($1024$) strings is equally likely. Since the probabilities must add up to $1$, each string has probability $\frac{1}{2^{10}}$. So for example the outcome HHHHHHHHHH is just as likely as the outcome HTTHHTHTHT. This may have an intuitively implausible feel, but it fits in very well with experiments.

Now let us assume that we will be happy only if we get exactly $3$ heads. To find the probability we will be happy, we count the number of strings that will make us happy. Suppose there are $k$ such strings. Then the probability we will be happy is $\frac{k}{2^{10}}$.

Now we need to find $k$. So we need to count the number of strings that have exactly $3$ H's. To do this, we find the number of ways to choose where the H's will occur. So we must choose $3$ places (from the $10$ available) for the H's to be.

We can choose $3$ objects from $10$ in $\binom{10}{3}$ ways. This number is called also by various other names, such as $C_3^{10}$, or ${}_{10}C_3$, or $C(10,3)$, and there are other names too. It is called a binomial coefficient, because it is the coefficient of $x^3$ when the expression $(1+x)^{10}$ is expanded.

There is a useful formula for the binomial coefficients. In general $$\binom{n}{r}=\frac{n!}{r!(n-r)!}.$$

In particular, $\binom{10}{3}=\frac{10!}{3!7!}$. This turns out to be $120$. So the probability of exactly $3$ heads in $10$ tosses is $\frac{120}{1024}$.

Remark: The idea can be substantially generalized. If we toss a coin $n$ times, and the probability of a head on any toss is $p$ (which need not be equal to $1/2$, the coin could be unfair), then the probability of exactly $k$ heads is $$\binom{n}{k}p^k(1-p)^{n-k}.$$ This probability model is called the Binomial distribution. It is of great practical importance, since it underlies all simple yes/no polling.

Video Transcript

For this exercise, we are told that coin is flipped 14 times and we know that it's a fair coin. So the probability of turning up tails is one half. We are asked to find the probability that more than six of the flips turn up tails. So let's define the random variable X as the number of tails Out of 14 flips. Now each flip can be viewed as a Bernoulli trial because each flip is independent from the other flips. So the outcome for one has no bearing on the others. The probability of success is always half. Now the number of successes in a series of Bernoulli trials is a binomial random variable. So here X is a binomial random variable and the binomial has two parameters, probability of success and and number of trials and so we want the probability of more than six tails Out of 14 flips. That is the probability that X is greater than six. Now for a binomial random variable, the probability mass function is given by this formula and choose X. That's peter the exponent X. Um so 1 -2 to the exponent and mind sex And X can take on any integer value from zero up to N. Now we can express this probability probability as one the probability that X Is at most six And then that would be 1- this summation. This isn't too bad to calculate by hand, but there's eight terms, they're seven or eight terms. So it's much more convenient to use software. Let's use Excel for this. So in Excel we can start by entering equals to start a calculation First we had 1- and then we want the probability that X is at most six. So we start to type binomial and this is the function that we want bynum dot d i s T. So we enter six successes. Number of trials is 14, Probability of success is .5. And then for a cumulative we select true because we want a cumulative probability We're looking for the probability that X is less than or equal to six and hit enter And we get .0.6047. So that is the answer. And that looks like it's the 4th multiple choice option in your question. Okay. I guess that's it. I hope this helped. And good luck in your studies.

What is the probability of flipping a coin 3 times and getting tails?

Answer: The probability of flipping a coin three times and getting 3 tails is 1/8.

What is the probability of getting no more than 3 heads when you flip a fair coin 5 times?

Considering a fair coin, after 5 flips, there are 25 = 32 different arrangements of heads and tails. Therefore, the probability of exactly 3 heads is 5/16.

What is the probability of getting 3 heads or less in 10 tosses of a fair coin?

So the probability of getting exactly three heads-- well, you get exactly three heads in 10 of the 32 equally likely possibilities. So you have a 5/16 chance of that happening.

What is the probability that when you flip a coin 3 times you will get 3 heads?

Correct answer: If you flip a coin, the chances of you getting heads is 1/2. This is true every time you flip the coin so if you flip it 3 times, the chances of you getting heads every time is 1/2 * 1/2 * 1/2, or 1/8.

When 5 fair coins are tossed what is the probability of getting more heads than tails?

What is the probability of flipping a coin and getting heads 11 times in a row?

What is the probability of getting exactly 5 heads and 5 tails in 10 flips?

What is the probability that we toss a coin 10 times and get at least 5 heads?