STATS-What is the probability of drawing a heart f

What is the probability of drawing a heart from a standard deck of cards?

Solution by Steps

step 1

Identify the total number of cards in a standard deck: A standard deck of cards has 52 cards

step 2

Determine the number of hearts in a standard deck: There are 4 suits in a deck of cards, and each suit has 13 cards. Therefore, there are 13 hearts in a deck

step 3

Calculate the probability: The probability

P

of drawing a heart from a standard deck of cards is the number of hearts divided by the total number of cards.

P = \frac{\text{Number of hearts}}{\text{Total number of cards}}

step 4

Substitute the numbers into the probability formula:

P = \frac{13}{52}

Answer

The probability of drawing a heart from a standard deck of cards is

\frac{13}{52}

or

\frac{1}{4}

.

Key Concept

Calculating the probability of a single event

Explanation

The probability of drawing a heart from a standard deck of cards is found by dividing the number of heart cards by the total number of cards in the deck.

Question 2 (15 marks) Nowadays, it is imperative for us to have a portable telecommunication device such as smart phone, iPad. A survey has been conducted to study the number of portable telecommunication devices owned by a person, and the results are as follows: \begin{tabular}{|l|l|l|l|l|} \hline \begin{tabular}{l} Number of portable telecommunication \\ devices owned (X) \end{tabular} & 1 & 2 & 3 & 4 \\ \hline Number of respondents & 32 & 40 & 12 & 16 \\ \hline \end{tabular} (a) Tabulate the probability distribution of the number of portable telecommunication devices owned by a person

(\mathrm{X})

. (b) Calculate the expected value and variance of the number of portable telecommunication devices owned by a person. (c) For an online shop, the probability of a customer buying an iPad is 0.1 . A random sample of 12 customers are selected. (i) Calculate the probability that at least 3 of them buy an iPad. (ii) Most likely, how many of them buy an iPad?

Solution by Steps

step 1

To tabulate the probability distribution, we first need to calculate the total number of respondents. This is done by adding the number of respondents for each number of devices owned

step 2

The total number of respondents is calculated as:

Total = 32 + 40 + 12 + 16

step 3

The probability distribution is found by dividing the number of respondents for each number of devices by the total number of respondents

step 4

The probabilities for each number of devices owned are:

P(X=1) = \frac{32}{Total}, P(X=2) = \frac{40}{Total}, P(X=3) = \frac{12}{Total}, P(X=4) = \frac{16}{Total}

Answer

The probability distribution is:

P(X=1) = \frac{32}{100}, P(X=2) = \frac{40}{100}, P(X=3) = \frac{12}{100}, P(X=4) = \frac{16}{100}

Key Concept

Probability distribution of a discrete random variable

Explanation

The probability distribution lists the probabilities of all possible values of a discrete random variable.

step 5

To calculate the expected value (mean) of X, we multiply each value of X by its corresponding probability and sum these products

step 6

The expected value is:

E(X) = 1 \cdot P(X=1) + 2 \cdot P(X=2) + 3 \cdot P(X=3) + 4 \cdot P(X=4)

step 7

To calculate the variance of X, we need to find the expected value of

X^2

and then use the formula

Var(X) = E(X^2) - [E(X)]^2

step 8

The expected value of

X^2

is:

E(X^2) = 1^2 \cdot P(X=1) + 2^2 \cdot P(X=2) + 3^2 \cdot P(X=3) + 4^2 \cdot P(X=4)

step 9

The variance is calculated using the expected values:

Var(X) = E(X^2) - [E(X)]^2

Answer

The expected value is

E(X) = 2.52

and the variance is

Var(X) = 1.2696

Key Concept

Expected value and variance of a discrete random variable

Explanation

The expected value is the average outcome if the experiment is repeated many times, while the variance measures the spread of the distribution.

step 10

To calculate the probability that at least 3 customers buy an iPad, we use the binomial probability formula

step 11

The binomial probability formula is:

P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}

where

n

is the number of trials,

k

is the number of successes, and

p

is the probability of success on a single trial

step 12

We need to sum the probabilities for

k = 3

to

k = 12

to find the probability of at least 3 customers buying an iPad

step 13

The probability is calculated as:

P(X \geq 3) = \sum_{k=3}^{12} \binom{12}{k} (0.1)^k (0.9)^{12-k}

Answer

The probability that at least 3 customers buy an iPad is approximately 0.0134

Key Concept

Binomial distribution and cumulative probability

Explanation

The cumulative probability of at least

k

successes in

n

binomial trials is the sum of the probabilities of

k

through

n

successes.

step 14

To find the most likely number of customers buying an iPad, we look for the value of

k

that maximizes the binomial probability formula for

k = 0

to

k = 12

step 15

The most likely number of customers, also known as the mode, is typically

np

for a binomial distribution when

np

is not an integer. When

np

is an integer, the mode is

np

or

np - 1

, whichever produces a higher probability

step 16

For our case,

np = 12 \cdot 0.1 = 1.2

, so we check the probabilities for

k = 1

and

k = 2

Answer

The most likely number of customers to buy an iPad is 1.

Key Concept

Mode of a binomial distribution

Explanation

The mode is the value of

k

for which the binomial probability is highest.

Question 1 (15 marks) (a) The management of a local University wants to study the workload of their students. A sample of 24 students were asked to indicate the average number of hours spent on doing assignments in a week. The followings are the results: \begin{tabular}{llllllllllll} 10 & 21 & 18 & 14 & 30 & 8 & 6 & 25 & 10 & 11 & 23 & 9 \\ 21 & 20 & 14 & 15 & 15 & 27 & 32 & 26 & 1 & 7 & 8 & 19 \end{tabular} (i) Calculate the mean and standard deviation. (ii) Find the

1^{\text {st }}

quartile, median and

3^{\text {rd }}

quartile. (iii) Comment on the skewness. State your reason. (b) In a canteen of the University, a lunch set is provided with a selling price of

\$ 50

. The number of lunch sets sold in a week

(\mathrm{X})

has a mean of 250 and a standard deviation of 20. Suppose the cost of running the canteen includes a fixed cost of

\$ 10000

per week and a variable cost of

\$ 15

per lunch set. (i) Express the weekly profit (P) in terms of

\mathrm{X}

. (ii) Hence, calculate the mean and standard deviation of the weekly profit.

AskSia

Plus