probability concepts

Probability: Concepts

Probability Concepts

We use three basic concepts or approaches to determine the probability of a given outcome. Which one we use depends on the experiment we're conducting -- what it is we're measuring.

1) Classical Probability Concept
If there are n equally likely possibilities
of which s are considered favourable or a successes,
the probability of a success is s / n.

Note: the outcomes of the experiment must all be EQUALLY LIKELY.

If we flip a coin and we want the probability of getting HEADS, or we toss a balanced die wanting the probability of rolling 5, we use the Classical probability concept. There is nothing to measure or observe since we know exactly how many equally likely outcomes there are. In the coin flip, the probability of flipping HEADS is ½ or 0.50 -- since there are 2 possible outcomes and HEADS has been labeled a success. For the balanced die, there's a 1 in 6 chance of rolling 5 since there is one success (5) and 6 equally likely possibilities.

2) Relative Frequency Concept
the probability of a success is the proportion
of times a success will occur in the long run,
based on a number of past observations.

If we want to know the probability that our home team will win their next hockey game, we use the Relative Frequency concept -- based on their past performance. If they won 7 of their last 10 games, the probability of a win would be 0.70. When survey results show that 64% of the respondents agree with the question, we say the probability of a majority voting for the proposal is 0.64.

3) Subjective Probability Concept
the likelihood of an event determined
by whatever information is available.

The Subjective Probability concept is more "homey" than formal. Subjective probabilities are commonly called educated guesses. If Aunt Martha's big toe hurts like hell the night before a rainy day, family members will predict stormy weather when they hear her complain. Sayings like "Red sky at night, sailor's delight" are probability statements from olden days. It means if the sky is red at sunset tonight, tomorrow will probably be a clear day.

We use all three methods in our work with probability distributions, however the Classical and Relative Frequency concepts are the most commonly applied.

Note: probabilities are positive proportions or fractions less than or equal to 1.

Examples

1) When we draw 1 card from a well-shuffled deck of 52, the probability of getting:

a) a black queen = 2 / 52 = 1 / 26
since there are 2 "successes" (black queens) in 52 equally likely choices.

b) a queen, king or ace of any suit = 12 / 52 = 3 / 13
since there are 12 "successes" (4 queens, 4 kings, 4 aces) in 52 cards.
Note: There are 3 "successes" in each of the 4 = 3 / 13.

c) a red card = 1 / 2 since half the deck is red.

d) a 5, 6, 7 or an 8 = 16 / 52 = 4 / 13 or 4 successes in each suit of 13 cards.

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Now we find the number of ways to succeed and the number of ways to perform
the experiment, using the Binomial theorem.

2) When we draw 3 cards from a well-shuffled deck of 52, the probability of getting:

a) 3 spades =

since we need 3 of 13 spades (successes) when we choose 3 of 52 cards.
The numerator, ₁₃C₃ , is the number of ways to succeed -- to get 3 of 13 spades.
The denominator, ₅₂C₃ , is b) 2 kings and a queen =

In the top we've got the number of ways to pick 2 of 4 kings AND 1 of 4 queens.
In the bottom is the number of ways to choose 3 cards from the deck of 52.

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3) If we draw one bead of 12 red, 20 white, 15 blue, and 3 black beads in a bag:

a) P(red) = 12 / 50 = 6 / 25

b) P(white or blue) = (20 + 15) / 50

c) P(neither white nor black) = (12 + 15) / 50 = 27 / 50
its just the sum of the red and blue (successes) over the total.

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4) The 8 possible outcomes of 3 coin flips are listed:

H H H	H H T	H T H	T H H
H T T	T H T	T T H	T T T

The probability of getting exactly:

a) 0 Heads = 1 / 8 b) 1 Head = 3 / 8 c) 2 Heads = 3 / 8 d) 3 Heads = 1 / 8.

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Practice

1) We draw 2 cards from a well-shuffled deck of 52, find the probability of getting:

a) 2 red cards? b) 2 aces? c) a 6 and a 7? d) 1 card of each color?

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2) When we roll a balanced die, what is the probability of rolling:

a) a 4? b) a 6 or 7? c) a number > 4? d) a number < 5?

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3) Ken and Barbie Cardshark play bridge with their friends the Pokerpals on Saturday night. Before the game, they draw lots for their chairs. What is the probabiltiy that Ken will not sit opposite Barbie?

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4) A box of 24 light bulbs includes 2 defective ones. If 2 bulbs are chosen at random,
what is the probability that:

a) neither is defective? b) one of them is defective? c) both of them are defective?

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5) Of a collector's 40 rare and ancient coins, 24 were minted in Rome, the rest in Alexandria.
If 4 coins are chosen at random, what is the probability that we get:

a) 2 from each city? b) one from Rome? c) all 4 from Alexandria?

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Solutions

1) We draw 2 cards from a well-shuffled deck of 52, find the probability of getting:

a) . b) .

c) . d)

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2) When we roll a balanced die, what is the probability of rolling:

a) 1 / 6 b) 2 / 6 = 1 / 3 c) 2 / 6 = 1 / 3 d) 4 / 6 = 2 / 3

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3) Once Barbie is seated, there are 3 seats left to fill. 1 seat is opposite her
and 2 are not, so the probability of NOT sitting opposite Barbie = 2 / 3.

4) A box of 24 light bulbs includes 2 defective ones. If 2 bulbs are chosen at random,
what is the probability that:

a) . b) . c) .

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5) Of a collector's 40 rare and ancient coins, 24 were minted in Rome, the rest in Alexandria.
If 4 coins are chosen at random, what is the probability that we get:

a) . b) . c) .

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