Seven coins are flipped what is the probability of getting fewer than three heads

We received a large number of responses of excellent quality.  

Ben from St Peter's followed the tree diagram and calculated out the answer:

Llewellyn from St Peter's and Diamor from Willington County Grammar School both observed an interesting pattern and expanded the answer to flipping ten coins:

Neeraj from Wilson School developed a generalization for different numbers of possible outcomes:

One student suggested how to calculate the number of desired outcomes:

Luke from Maidstone Grammar School went further to investigate the next part of the question:

If a ball is selected and then replaced the probability of picking a red ball or a green ball is the same every time. When 3 balls are picked with replacement the probability of getting at least one green is

When the balls are not replaced the probability of getting at least one green is still 1-(the probability of getting 3 reds). In each draw the probability of drawing a red ball is $\frac{\text{the number of red balls}}{\text{the total number of balls}}$

Helen from Stroud finished up the problem:

Well done to everyone.

Nurcan M.

Suppose Mike is playing a coin game as follows. First he throws three fair coins into the air (First Toss). For each “heads”, he wins a dollar. Then, he tosses the three coins again (Second Toss). For each “heads” on this second toss, he also wins a dollar. (So he can win anywhere from $0 to $6 in one play of the game.)

a)What is the probability of tossing three heads on the first toss?

b)What is the probability of tossing fewer than 3 heads on the first toss?

c)What is the probability of tossing 0 heads on the first toss?

d)Suppose Mike tosses 0 heads on the first toss. What is the probability of doing better than that on the second toss?

e)Suppose Mike tosses 0 heads on the first toss and then curses the coins (“You stupid coins!”) What is the probability of doing better on the second toss?

f)Suppose Mike tosses 3 heads on the first toss. What is the probability of doing worse than that on the second toss?

g)Suppose Mike tosses 3 heads on the first toss, and then praises the coins (“Good coins!”) What is the probability of doing worse on the second toss?

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.

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!