What is Probability?

Let’s start by understanding what probability is. In simple terms, probability is a way of expressing the likelihood that a specific event will occur. It’s expressed as a number between 0 (the event will definitely not happen) and 1 (the event will definitely happen). For example, flipping a coin has a probability of 0.5 (or 1/2) that it will land on heads, because there are 2 equally likely outcomes (heads and tails), and one of them is ‘heads’.

Independent Events

Now, let’s talk about ‘independent events’. Two events are considered independent if the occurrence of one event does not affect the probability of the occurrence of the other. For example, if you roll a dice and flip a coin, the outcome of the dice roll does not affect the outcome of the coin flip. These are independent events.

The key rule to remember for independent events is:

If A and B are independent events, the probability of both events occurring is the product of their individual probabilities.

So, if Event A (rolling a 3 on a dice) has a probability of 1/6, and Event B (flipping a heads on a coin) has a probability of 1/2, the probability of both events occurring is:

P(A and B) = P(A) x P(B) = 1/6 x 1/2 = 1/12

Conditional Probability

Next, we have ‘conditional probability’. This is the probability of an event occurring, given that another event has already occurred. For example, if we have a bag of 5 red balls and 5 blue balls, and we take out a red ball without replacing it, the probability of drawing another red ball has changed. These are dependent events, and we use conditional probability to figure out the new probability.

The key rule for conditional probability is:

The probability of Event A given Event B is the probability of Event A and B divided by the probability of Event B.

So, if Event A is drawing a red ball (with 5 red and 5 blue balls in the bag), and Event B is drawing another red ball (with now 4 red and 5 blue balls in the bag), the probability of both events occurring is:

P(A and B) = P(A) x P(B given A) = 5/10 x 4/9 = 2/9

And the probability of Event B given Event A is:

P(B given A) = P(A and B) / P(A) = 2/9 / 5/10 = 4/9

Practice Questions

Now let’s try some practice questions. Remember, the best way to understand probability is to practice!

Easy Questions

  1. What is the probability of rolling a 6 on a dice?
  2. What is the probability of flipping a tails on a coin?
  3. If you roll a dice twice, what is the probability of rolling a 6 both times?
  4. If you flip a coin twice, what is the probability of flipping heads both times?
  5. If you have a bag of 5 red balls and 5 blue balls, what is the probability of drawing a red ball?
  6. If you draw a red ball from the bag (and do not replace it), what is the probability of drawing another red ball?
  7. What is the probability of drawing a blue ball from the bag after drawing a red ball?
  8. If you flip a coin and roll a dice, what is the probability of flipping a heads and rolling a 3?
  9. If you roll a dice twice, what is the probability of rolling a 2 and then a 4?
  10. If you flip a coin twice, what is the probability of flipping a tails and then a heads?

Medium Questions

  1. What is the probability of rolling a 1, 2, or 3 on a dice?
  2. What is the probability of flipping a heads on a coin and rolling a 1, 2, or 3 on a dice?
  3. If you have a bag of 3 red balls and 7 blue balls, what is the probability of drawing a red ball?
  4. If you draw a red ball from the bag (and do not replace it), what is the probability of drawing another red ball?
  5. What is the probability of drawing a blue ball from the bag after drawing a red ball?
  6. If you flip a coin and roll a dice, what is the probability of flipping a tails and rolling a 5?
  7. If you roll a dice twice, what is the probability of rolling a 1 and then a 6?
  8. If you flip a coin twice, what is the probability of flipping a heads and then a tails?
  9. What is the probability of rolling a 3 on a dice, given that you’ve already rolled a 3?
  10. What is the probability of flipping a tails on a coin, given that you’ve already flipped a heads?

Hard Questions

  1. If you have a bag of 4 red balls, 3 blue balls, and 3 green balls, what is the probability of drawing a red ball?
  2. If you draw a red ball from the bag (and do not replace it), what is the probability of drawing another red ball?
  3. What is the probability of drawing a blue ball from the bag after drawing a red ball?
  4. What is the probability of drawing a green ball from the bag after drawing a red and a blue ball?
  5. If you flip a coin and roll a dice, what is the probability of flipping a heads and rolling a 6?
  6. If you roll a dice twice, what is the probability of rolling a 6 and then a 1?
  7. If you flip a coin twice, what is the probability of flipping a heads and then a tails?
  8. What is the probability of rolling a 2 on a dice, given that you’ve already rolled a 6?
  9. What is the probability of flipping a heads on a coin, given that you’ve already flipped a tails?
  10. If you have a bag of 3 red balls, 2 blue balls, and 5 green balls, what is the probability of drawing a green ball, given that you’ve already drawn a red ball?

The answers to the questions are provided at the end. This will allow you to check your understanding and get feedback on your progress.

Answers

Easy Questions

  1. 1/6
  2. 1/2
  3. 1/36
  4. 1/4
  5. 1/2
  6. 4/9
  7. 5/9
  8. 1/12
  9. 1/36
  10. 1/4

Medium Questions

  1. 1/2
  2. 1/4
  3. 3/10
  4. 2/9
  5. 7/9
  6. 1/12
  7. 1/36
  8. 1/4
  9. 1/6
  10. 1/2

Hard Questions

  1. 4/10 = 2/5
  2. 3/9 = 1/3
  3. 3/9 = 1/3
  4. 3/8
  5. 1/12
  6. 1/36
  7. 1/4
  8. 1/6
  9. 1/2
  10. 5/9

Remember, understanding probability takes practice, so keep trying these problems until you feel comfortable with them. Good luck!