Probability

[mashshare]

Probability is a section of mathematics that is all about calculating the possibility of a given event’s occurrence, which is expressed as a number between 1 and 0.

Simply put:

Probability =  the number of ways of achieving success
 the total number of possible outcomes

An event that has a probability of 1 can be assumed as a certainty: for instance, when a coin is tossed, the result could be either “heads” or “tails” is 1 as there are no other options in this situation (assuming the coin lands flat). An event that has a probability of 0.5 can be thought of having equal odds of happening or not happening. Let’s say, the possibility of a coin toss that results in “heads” is .5 as the toss is equally as likely to result in “tails.”

An event that has a probability of 0 can be considered to be an impossible occurrence. For instance, the chance of the coin landing (flat) without either side facing up is 0, as here, either “heads” or “tails” should happen facing up. A little ironic, but the probability theory is applicable for figuring out precise calculations and quantifying uncertain measures of random events.

Simply put, probability is expressed in mathematical terms as: the number of occurrences of a targeted event divided by the number of occurrences plus the number of failures of occurrences (this adds up to the total of possible outcomes):

p(a) = p(a)/[p(a) + p(b)]

When it comes to calculating probabilities in a condition like a coin toss is pretty simple since the outcomes are more often than not mutually exclusive: either one event or the other must occur. Note that every coin toss is an independent occurrence, and the outcome of one trial does not have any effect on subsequent ones. No matter how many successive times one side faces up while landing, the chances of it being at the same position in the next toss is always 0.5 (50-50). The entire incorrect notion that a certain number of consecutive results (six “heads” for example) increases the chances that the next toss will result in a “tails” is called as the gambler’s fallacy. This theory has lead to the downfall of many intelligent minds.

Interestingly, probability theory was discovered in the 17th century, when two French mathematicians, Blaise Pascal and Pierre de Fermat were carrying out a correspondence and discussing mathematical problems about games of chance. Modern applications of probability theory do go through many human inquiries, and also incorporate various aspects of computer programming, weather prediction, astrophysics, music, and medicine.

Calculating Probability

For calculating probability, one needs to understand the number of possible options or outcomes along with how many right combinations we can have. Let’s begin with calculating the probability of throwing dice to see how it works.

Firstly, we all are aware that a die (plural: dice) consists of a total of 6 possible outcomes. When we play, we can roll a 1, 2, 3, 4, 5, or 6. Secondly, we need to understand how many choices we have. Every time we roll, we might get one of the numbers. We cannot obviously roll and get two different numbers with one die. So, we have only 1 number of choices. We get a probability of 1/6 after using our formula for probability.

Our chances of rolling any one of the numbers are always 1/6. The likelihood of rolling a 2 is 1/6, of rolling a 3 is also 1/6, and so on.

Let’s take a look at another problem. Let’s assume that we have a bag filled with apples and oranges. In this situation, we need to calculate the probability of picking an apple from the bag. The first thing we need to know is the actual number of apples present in the bag as that helps us figure out the number of ‘correct’ choices, which are nothing but the number of our possible choices.

We also need to keep in mind the overall number of fruits present in the bag as that knowledge helps us with understanding the total number of choices or the total number of options we have. The person who has the grab bag would be the right one to tell us if there are 10 apples and 20 oranges inside the bag. So, what are our chances of selecting an apple? We have a total of 10 apples, one of which we want, and a total of 30 fruits to pick from.

For picking an apple, our probability is 1/3. Remember that the chances of picking an apple from the grab bag are higher when we compare it with our probability of rolling a number on a die. Our chances of picking up an apple are higher than rolling a particular number.

Single Event

Example

Let’s say there is a bag filled with balls in colours red, blue, green and orange. All the balls get picked out and replaced. Peter did this 1000 times and got the following results:

  • Number of blue balls picked out: 300
  • Number of red balls: 200
  • Number of green balls: 450
  • Number of orange balls: 50
  1. a) What is the probability of picking a green ball?

For each set of 1000 balls picked out, 450 are green. Therefore P(green) = 450/1000 = 0.45

  1. b) If there are 100 balls in the bag, how many of them are more expected to be green?

As per the experiment, 450 out of 1000 balls are green. Thus, 45 are green (using ratios) out of 100 balls.

Multiple Events

Independent and Dependent Events

Suppose there’s a probability of two events happening at the same time. For instance, we land up throwing 2 dice and think about the probability of both being 6s.

In this situation, two events could be called independent if the result of one of the events does not affect the outcome of another. Case in point, if we throw two dice at the same time, no matter what we get with the first one- it’s still 1/6, the chances of getting a 6 on the second die is the same.

On the contrary, presume we have a bag that has 2 red and 2 pink balls. If we select 2 balls from the bag, the chances that the second is pink are totally dependent on the colour of the first ball that was picked. If the colour of the first ball was pink, then there will be 1 pink and 2 red balls in the bag when we pick the second ball. So, this way, the probability of having a pink ball is 1/3. But, if the first ball we picked was red, then there would be only 1 red and 2 pink balls left. So, the probability of the second ball being pink is 2/3. In such a scenario, when the probability of one event depends on another, the events are dependent.

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