Chance and Probability

Sometimes it happens that during rainy season, you carry a raincoat every day and it does not rain for many days. However, by chance, one day you forget to take the raincoat and it rains heavily on that day.

Sometimes it so happens that a student prepares 4 chapters out of 5, very well for a test. But a major question is asked from the chapter that she left unprepared. Everyone knows that a particular train runs in time but the day you reach well in time it is late!

You face a lot of situations such as these where you take a chance and it does not go the way you want it to. Can you give some more examples? These are examples where the chances of a certain thing happening or not happening are not equal. The chances of the train being in time or being late are not the same. When you buy a ticket which is wait listed, you do take a chance. You hope that it might get confirmed by the time you travel.

We however, consider here certain experiments whose results have an equal chance of occurring.

Getting a result

Try these

1. If you try to start a scooter, what are the possible outcomes?

Solution :

Instructions

2. When a die is thrown, what are the possible outcomes?

Solution :

Instructions

When you spin the wheel shown, what are the possible outcomes? (Outcome here means the sector at which the pointer stops).

Think, Discuss and Write

In throwing a die:

Does the first player have a greater chance of getting a six? NoYes

Would the player who played after him have a lesser chance of getting a six? NoYes

Suppose the second player got a six. Does it mean that the third player would not have a chance of getting a six? NoYes

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Equally likely outcomes:

A coin is tossed several times and the number of times we get head or tail is noted. Let us look at the result sheet where we keep on increasing the tosses:

Observe that as you increase the number of tosses more and more, the number of heads and the number of tails come closer and closer to each other. This could also be done with a die, when tossed a large number of times. Number of each of the six outcomes become almost equal to each other.

In such cases, we may say that the different outcomes of the experiment are equally likely. This means that each of the outcomes has the same chance of occurring.

What are Probabilities

A probability is a number between 0 and 1 which describes the likelihood of a certain event. A probability of 0 means that something is impossible; a probability of 1 means that something is certain.

For example, it is impossiblecertain that you will meet a real life dragon, and it is certainimpossible that the sun will rise tomorrow. The probability of a coin landing heads is exactly 50%100%.

The probability of rolling a 6 on a die, or picking a particular suit from a deck of cards is lessmore than 0.5 – which means unlikely. The probability of a good football team winning a match, or of a train arriving on time is moreless than 0.5 – which means likely.

Now drag the following events into the correct order, from likely to unlikely:

Instructions

You throw a die and it lands on 6.

Penguins live on the North Pole.

It’s going to rain in November.

A baby will be born in China today.

You buy a lottery ticket and win the Jackpot .

A newborn baby will be a girl .

We often use probabilities and likelihoods in everyday life, usually without thinking about it. What is the chance of rain tomorrow? How likely is it that I will miss the bus? What is the probability I will win this game?

Tossing a (fair) coin has two possible outcomes, heads and tails, which are both equally likely. According to the equation above, the probability of a coin landing heads must be 12 = , or 50%.

Note that this probability is in between 0 and 1, even though only one of the outcomes can actually happen. But probabilities have very little to do with actual results: if we toss a coin many times we know that approximately halfexactly halfallnone of the results are heads – but we have no way of predicting exactly which tosses landed heads.

Even events with tiny probabilities (like winning the lottery ) can still happen – and they do happen all the time (but to a very small proportion of the people who participate).

Probabilities also depend on how much each of us knows about the event. For example, you might estimate that the chance of rain today is about 70%, while a meteorologist with detailed weather data might say the chance of rain is 64.2%.

Or suppose that I toss a coin and cover it up with my hands – the probability of tails is 50%. Now I peek at the result, but don’t tell you. I know for certain what has happened, but for you the probability is still 50%100%not 50%.

There are many different ways to think about probabilities, but in practice they often give the same results:

The classical probability of landing heads is the proportion of possible outcomes that are heads.

The frequentist probability is the proportion of heads we get if we toss the coin many times.

The subjectivist probability tells us how strongly we believe that the coin will land heads.

Remember that while probabilities are great for estimating and forecasting, we can never tell what actually will happen.

Continue

Linking chances to probability

Consider the experiment of tossing a coin once. What are the outcomes? There are only two outcomes – Head or Tail. Both the outcomes are equally likelyunequally likely.

Likelihood of getting a head is one out of two outcomes, i.e., 1224

In other words, we say that the probability of getting a head = 12.

What is the probability of getting a tail?

Now take the example of throwing a die marked with 1, 2, 3, 4, 5, 6 on its faces (one number on one face). If you throw it once, what are the outcomes? The outcomes are: , , , , , .

Thus, there are equally likely outcomes.

What is the probability of getting the outcome ‘2’?

(ii) What is the probability of getting the number 7?

Instructions

(iii)What is the probability of getting a number 1 through 6?

Instructions

Outcomes as events

Each outcome of an experiment or a collection of outcomes make an "event". For example: in the experiment of tossing a coin, getting a Head is an event and getting a Tail is also an event.

In case of throwing a die, getting each of the outcomes 1, 2, 3, 4, 5 or 6 is an event.

Is getting an even number an event? Since an even number could be 2, 4 or 6, getting an even number is also an event. What will be the probability of getting an even number?

It is 36 i.e. 12 ← Number of outcomes that make the event

6 ← Total number of outcomes of the experiment.

Example 3: A bag has 4 red balls and 2 yellow balls. (The balls are identical in all respects other than colour). A ball is drawn from the bag without looking into the bag. What is probability of getting a red ball? Is it more or less than getting a yellow ball?

Solution: There are a total numer of outcomes of the event. (= 4 + 2 )

Getting a red ball consists of outcomes.

Therefore, the probability of getting a red ball = 4626 i.e. equal to 2313.

In the same way the probability of getting a yellow ball = 2636 i.e. equal to 1323.

Therefore, the probability of getting a red ball is moreless than that of getting a yellow ball.

Try These

Suppose you spin the wheel

1. (i) List the number of outcomes of getting a green sector and not getting a green sector on this wheel.

(ii) Find the probability of getting a green sector.

(iii) Find the probability of not getting a green sector.

Chance and probability related to real life

We talked about the chance that it rains just on the day when we do not carry a rain coat. What could you say about the chance in terms of probability? Could it be one in 10 days during a rainy season? The probability that it rains is then 110.

The probability that it does not rain = 910110. (Assuming raining or not raining on a day are equally likely).

The use of probability is made in various cases in real life.