Probability

Probability

003: Engineering Principles

​Lesson Objectives

To be able to:

• Understand what an unconditional probability is

• Calculate unconditional probabilities

• Understand what a conditional probability is

• Calculate conditional probabilities

• Probability is the measure of the likelihood that an event will occur.

• Probabilities can be stated as decimals, fractions, and percentages

• For decimals and fractions, probability can be expressed on a scale that ranges from 0 to 1

​

​Introduction

Multiple Choice

A probability closer to zero means the event is...

• Likely to occur

• Unlikely to occur

Multiple Choice

A probability closer to one means the event is...

• Likely to occur

• Unlikely to occur

The value of probability, P, is defined simply as:​

​Introduction

In the case of flipping a coin to get tails:

​Introduction

1

2

=

1 wanted outcome (tails)

2 possible outcomes (heads and tails)

= 0.5 = 50%

In the case of rolling a dice to get 6:

​Introduction

1

6

=

1 wanted outcome (6)

6 possible outcomes (1, 2, 3, 4, 5, 6)

= 0.167 = 16.7%

Math Response

What is the probability of rolling a dice to get an odd number?

Type answer here

Correct Answer: \frac{1}{2}

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Math Response

What is the probability of rolling a dice to get a number greater than 4?

Type answer here

Correct Answer: \frac{1}{3}

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Math Response

What is the probability of rolling a dice to get a number greater than 1?

Type answer here

Correct Answer: \frac{5}{6}

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Unconditional probability is when the probability of one event occurring does not depend on the probability of another event occurring - in other words, both events are independent of each other.

​Unconditional Probability

• The probability of flipping a coin twice to get heads twice - the probability of getting heads on the second coin flip is independent of the outcome of the first coin flip, and therefore will always be 0.5.

• The probability of rolling two dices to get two sixes - the probability of getting six on one dice is independent of the outcome of rolling the other dice, and therefore will always be 0.167

​Examples of Unconditional Probability

Example 1) Calculate probability of flipping a coin twice to get heads twice

​Calculating Unconditional Probability

1

2

P_(heads) =

Probability of two heads = P_(heads) x P_(heads)

= 0.5

Example 2) Calculate probability of rolling two dices to get two sixes

​Calculating Unconditional Probability

1

6

P_(six) =

Probability of two sixes = P_(six) x P_(six)

= 0.167

Math Response

Calculate the probability flipping a coin three times and getting tails all three times.

Type answer here

Correct Answer: \frac{1}{8}

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Math Response

A coin is tossed four times, what is the probability of the tosses coming up in the order HTHH?

Type answer here

Correct Answer: \frac{1}{16}

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Math Response

Calculate the probability of rolling an even number twice in two rolls of a dice.

Type answer here

Correct Answer: \frac{1}{4}

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Math Response

A dice is thrown, what is the probability that the number obtained is a prime number.

Type answer here

Correct Answer: \frac{1}{2}

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Math Response

Calculate the probability of rolling a four OR five on two rolls of a dice.

Type answer here

Correct Answer: \frac{1}{9}

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Math Response

Calculate the probability of rolling a four on the first throw of a dice and a five on the second throw of a dice.

Type answer here

Correct Answer: \frac{1}{36}

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Math Response

Calculate the probability of rolling a number greater than 2 on two successive rolls of a dice.

Type answer here

Correct Answer: \frac{4}{9}

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Math Response

If a dice is thrown twice, what is the probability of not getting a one on either throw?

Type answer here

Correct Answer: \frac{25}{36}

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Math Response

Calculate the probability of rolling a dice and getting three sixes in succession.

Type answer here

Correct Answer: \frac{1}{216}

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Math Response

Calculate the probability of rolling two dies and getting a score of three in total.

Type answer here

Correct Answer: \frac{1}{36}

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Conditional probability is when the probability of one event occurring does depend on the probability of another event occurring - in other words, both events are dependent of each other.

In such cases, the probability of the second event occurring will be different to the probability of the first event occurring (the probability of the second event can be lower or higher), which needs to be factored into the probability calculation.

​Conditional Probability

The probability of drawing two hearts from a deck of playing cards -

• The probability of drawing a hearts from a deck of cards is 13/52 or 25%; assuming the card drawn first is a hearts and is not returned to the deck, then the probability of drawing a hearts from the deck of cards is now 12/51 or 23.5%, which is lower than the first draw.

• Alternatively, assuming the card drawn first is not a hearts and is not returned to the deck, then the probability of drawing a hearts from the deck now 13/51 or 25.5%, which is higher than the probability on the first draw.

​Examples of Conditional Probability

Example 1) Calculate probability of drawing two hearts from a deck of playing cards.

​Calculating Conditional Probability

13

52

1st draw: P_(hearts1) =

Probability of two hearts = P_(hearts1) x P_(hearts2)

2nd draw: P_(hearts2) =

12

51

Example 2) Calculate probability of drawing two red cards from a deck of playing cards.

​Calculating Conditional Probability

26

52

1st draw: P_(red1) =

Probability of red cards = P_(red1) x P_(red2)

2nd draw: P_(red2) =

25

51

Math Response

You are drawing two cards from a deck of playing cards. Calculate the probability of drawing two kings in succession. Assume the first card is not placed back into the deck.

Type answer here

Correct Answer: \frac{1}{221}

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Math Response

You are drawing two cards from a deck of playing cards. Calculate the probability of drawing a red card on the first draw and a black card on the second draw. Assume the first card is not placed back into the deck.

Type answer here

Correct Answer: \frac{13}{51}

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Math Response

You are drawing two cards from a deck of playing cards. Calculate the probability of drawing a diamond in the first draw and a clubs in the second draw. Assume the first card is not placed back into the deck.

Type answer here

Correct Answer: \frac{13}{204}

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Math Response

You are drawing two cards from a deck of playing cards. Calculate the probability of drawing an ace of spades in the first draw and an ace of hearts in the second draw. Assume the first card is not placed back into the deck.

Type answer here

Correct Answer: \frac{1}{2652}

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Math Response

You are drawing two cards from a deck of playing cards. Calculate the probability of drawing a king or queen in succession. Assume the first card is not placed back into the deck.

Type answer here

Correct Answer: \frac{14}{663}

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Math Response

You are drawing three cards from a deck of playing cards. Calculate the probability of drawing three spades in succession. Assume the cards being picked are not placed back into the deck.

Type answer here

Correct Answer: \frac{11}{850}

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Math Response

You are drawing three cards from a deck of playing cards. Calculate the probability of drawing three kings in succession. Assume the cards being picked are not placed back into the deck.

Type answer here

Correct Answer: \frac{1}{5525}

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