Question

Probabilities of picking cards and rolling dice. (a) What is the probability of drawing either a queen or a heart in a normal deck of 52 cards? (b) What is the probability \(P\) of getting three 7 's and two 4 's on five independent rolls of a die?

Short answer

The probability of drawing either a queen or a heart is \(\frac{4}{13}\). The probability of rolling three 4’s and two 5’s is 0.0013.

Step-by-step solution

Identify total cards in the deck

A standard deck has 52 cards in total.

Calculate the probability of drawing a queen

There are 4 queens in a deck. Thus, the probability of drawing a queen is \(\frac{4}{52} = \frac{1}{13}\).

Calculate the probability of drawing a heart

There are 13 hearts in a deck. Thus, the probability of drawing a heart is \(\frac{13}{52} = \frac{1}{4}\).

Calculate the overlap (queens that are also hearts)

There is 1 queen of hearts in the deck. Thus, the probability of drawing the queen of hearts is \(\frac{1}{52}\).

Use the Inclusion-Exclusion Principle

Probability of drawing either a queen or a heart is given by combining the probabilities from Steps 2 and 3 and subtracting the overlap from Step 4. \[P(\text{Queen or Heart}) = P(\text{Queen}) + P(\text{Heart}) - P(\text{Queen of Hearts}) = \frac{1}{13} + \frac{1}{4} - \frac{1}{52}\]\[(\frac{4}{52} + \frac{13}{52} - \frac{1}{52}) = \frac{16}{52} = \frac{4}{13}\]

Calculate the probability for rolling dice (a combination problem)

Use the combination formula to determine the number of ways to get three 7s and two 4s out of five rolls. This will be a binomial coefficient: \(\binom{5}{3}\)

Use the binomial probability formula

The formula for binomial probability is: \[P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\] Given a die roll,the probability of getting a 7 is 0 (since die only have faces 1-6) so let’s change the problem to realistic numbers such as getting three 4s and two 5s.With six-sided die \(\frac{1}{6}\) for each event.Apply the formula: \(P(\text{three 4s, two 5s from 5 rolls}) = \binom{5}{3} (\frac{1}{6})^3 (\frac{1}{6})^2=10\cdot(\frac{1}{6})^5=\frac{10}{7776}Approx0.0013\)

Key concepts

deck of cards probability

Let's start with calculating the probability using a deck of cards. A standard deck has 52 cards, divided into 4 suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards.
In this context, we want to find the probability of drawing either a queen or a heart.
Step-by-step:
- There are 4 queens in the deck, so the probability of drawing a queen is \(\frac{4}{52} = \frac{1}{13}\).
- There are 13 hearts in the deck, so the probability of drawing a heart is \(\frac{13}{52} = \frac{1}{4}\).

We also need to account for the overlap, which is the queen of hearts. There's just 1 queen of hearts, so the probability is \(\frac{1}{52}\).
Now, applying the Inclusion-Exclusion Principle:
\[P(\text{Queen or Heart}) = P(\text{Queen}) + P(\text{Heart}) - P(\text{Queen of Hearts}) = \frac{1}{13} + \frac{1}{4} - \frac{1}{52}\text{.}\]
This results in:
\[\frac{4}{52} + \frac{13}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}\].

Thus, the probability of drawing either a queen or a heart from a standard deck of cards is \(\frac{4}{13}\).

binomial coefficient

The binomial coefficient is crucial for calculating combinations in probability problems. It denotes the number of ways to choose \(k\) successes in \(n\) independent trials.

For example, let's say we are rolling a die and need to calculate the probability of getting three 4s and two 5s in five rolls.
We use the binomial coefficient, notated as \(\binom{n}{k}\).
The formula for this is:
\(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).

In this case, we have \(n = 5\) and \(k = 3\), so:
\[\binom{5}{3} = \frac{5!}{3!2!} = \frac{120}{6 \times 2} = 10\].
There are 10 possible combinations to get three 4s and two 5s in five rolls of a die.
This helps us determine the precise arrangement possibilities.

inclusion-exclusion principle

The Inclusion-Exclusion Principle is used to find the probability of at least one of several events happening.

In our card example, we used it to determine the probability of drawing either a queen or a heart.
The principle ensures we don’t double-count outcomes that fall into multiple categories.

For instance, if we calculate the probability of a queen and a heart separately and just add them, we count the queen of hearts twice. The formula corrects for this overcounting.
Mathematically, it is expressed as:
\[P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)\].

Replacing with the values from our card example:
\(\frac{1}{13} + \frac{1}{4} - \frac{1}{52}\).
This principle is very useful in set theory and probability to avoid errors by overcounting intersections of sets/events.

rolling dice probability

Probability problems involving dice are common and can be fascinating! A standard six-sided die has faces numbered from 1 to 6. The probability of any specific number is thus \(\frac{1}{6}\).

To calculate the probability of rolling a specific combination, such as three 4s and two 5s in five rolls, we use the binomial probability formula:
\[P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\].
Here, \(n = 5\) (five rolls), and we want \(k = 3\) (three 4s) with \(p = \frac{1}{6}\) (probability for any face).
So, we also have \(\binom{5}{3} = 10\).
Therefore:
\[P(\text{three 4s and two 5s}) = 10 \times \frac{1}{6}^5 = 10 \times \frac{1}{7776} \text{, which simplifies to approx } 0.0013.\]

This kind of problem illustrates the application of both combinatorial calculations and probability theory. It helps us understand the likelihood of specific outcomes in sequential trials.