50% (i.e. 1/2)

What is the chance that tails will appear?

50% (i.e. 1/2)

Example 2:

Toss a dice. What is the probability that an even number will appear?

Solution:

The six numbers on a dice are 1, 2, 3, 4, 5, 6.

On the dice, 2, 4 and 6 are the even numbers.

So, the probability an even number will fall is 3/6 i.e. ½

Example 3:

In a bag, there are 3 green and 4 red marbles. Take out any two marbles out of the bag. What is the probability the two marbles are green-colored?

**Solution:** Bag contains 7 marbles. 2 marbles are taken out. 2 out of 7 can be taken out in ^{7}c_{2} ways i.e. 21 ways.

(Recall that r things can be selected out of n things in ^{n}c_{r} ways from combinations)

2 green out of 3 green marbles can be taken out in ^{3}c_{2} ways i.e. 3 ways.

The probability is therefore: ^{3}/_{21} = ^{1}/_{7}

From the three examples above, we can define probability as follows:

If E denotes an event, then the probability Event E will happen is:

**P (E) = (Outcomes favorable for E)/ (Total number of outcomes)**

The table below summarizes the salient points in the 3 examples above.

So, try a definition for Probability? Here we go:

Let E denote an Event and P (E) the probability that Event E will happen. Then,

**P (E) = ^{(Number of outcomes favorable for E)}/_{(Total number of outcomes)}**

1. 3 heads

2. At least 3 heads.

3. At most 2 heads.

4. 3 heads, exactly next to each other.

P (E) =

First find the total number of outcomes when 5 coins are tossed.

It is

(If 3 coins are tossed, the total number of outcomes is

So, 3 coins give

Likewise, 5 coins give

Now, how many outcomes show 3 heads out of 32?

The dots indicate Tails.

Tabulating is difficult.

You must use an easier way to find favorable outcomes, with a formula probably.

And it is combinations formula:

We need 3 Heads. Which 3 coins must show the required 3 Heads?

Any 3 coins. So, select any 3 coins out of the 5 coins. In how many ways?

Yes, in

So, the probability 3 Heads will appear is =

At least 3 heads means minimum 3 heads i.e.

3 or more heads i.e. 3 or 4 or 5 heads.

So, find probability of the following events:

(Remember one rule: Write

At most means maximum 2 heads i.e. up to 2 heads, i.e. less than or equal to 2 heads.

So, find probability of the following events:

2 Heads or 1 Heads or No Heads.

So, P (At most 2 Heads) = P (2H) + P (1H) + P (0H)

1^{st} coin |
2^{nd} coin |
3^{rd} coin |
4^{th} coin |
5^{th} coin |
---|---|---|---|---|

H | H | H | . | . |

. | H | H | H | . |

. | . | H | H | H |

We must tabulate the outcomes that give 3 Heads as it is a special case.

P (3 Heads, next to each other) = 3/32

(Note: In the first row, 5th coin cannot show a Heads, for we need only 3 Heads and the series 5th, 1st and 2nd coin is not acceptable for Heads to fall next to each other).