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Here are a few recommended readings before getting started with this lesson.
Kevin plans to invest his money in a cryptocurrency. He is choosing between two cryptocurrencies — Coin Q and Coin R. The table shows the probabilities of the events over the next week.
Lose $25 | Gain $10 | Gain $50 | |
---|---|---|---|
Coin Q | 0.40 | 0.20 | 0.40 |
Coin R | 0.10 | 0.75 | 0.15 |
Which cryptocurrency should Kevin choose? Justify the answer.
Trial | Number of points to the left of (2,2) | Number of points to the right of (2,2) |
---|---|---|
1 | 8 | 2 |
2 | 7 | 3 |
3 | 9 | 1 |
4 | 7 | 3 |
5 | 6 | 4 |
6 | 6 | 4 |
7 | 7 | 3 |
8 | 9 | 1 |
9 | 6 | 4 |
10 | 7 | 3 |
Trial | NL | NR |
---|---|---|
1 | 8 | 2 |
2 | 7 | 3 |
3 | 9 | 1 |
4 | 7 | 3 |
5 | 6 | 4 |
6 | 6 | 4 |
7 | 7 | 3 |
8 | 9 | 1 |
9 | 6 | 4 |
10 | 7 | 3 |
Consider a square whose sides have a length of 5 units. The applet shows 100 random points inside and a unit inside the square. Move the unit square to see how the percentage of the random numbers inside the unit square changes.
Help Maya estimate the area under the curve.
Trial | Number of Points Under the Graph | Number of Points Above the Graph |
---|---|---|
1 | 35 | 65 |
2 | 39 | 61 |
3 | 34 | 66 |
4 | 33 | 67 |
5 | 31 | 69 |
6 | 28 | 72 |
7 | 27 | 73 |
8 | 37 | 63 |
9 | 33 | 67 |
10 | 35 | 65 |
Trial | NU | NA |
---|---|---|
1 | 35 | 65 |
2 | 39 | 61 |
3 | 34 | 66 |
4 | 33 | 67 |
5 | 31 | 69 |
6 | 28 | 72 |
7 | 27 | 73 |
8 | 37 | 63 |
9 | 33 | 67 |
10 | 35 | 65 |
Substitute values
Vincenzo and his friends decide to play a game. They roll a die up to 60 times. They have to pay 2 coins if 1, 2, 3, or 4 appears. They get 5 coins if 5 or 6 appears.
Trial | Number of Coins |
---|---|
1 | 27 |
2 | -1 |
3 | 41 |
4 | 34 |
5 | 6 |
6 | 69 |
7 | -8 |
8 | -8 |
9 | 34 |
10 | 27 |
Average | 22.1 |
Multiply
Add and subtract terms
Trial | Number of Coins |
---|---|
1 | 27 |
2 | -1 |
3 | 41 |
4 | 34 |
5 | 6 |
6 | 69 |
7 | -8 |
8 | -8 |
9 | 34 |
10 | 27 |
Add and subtract terms
Calculate quotient
Under these conditions, the number of coins Vincenzo can get will be equal to 20.
Multiply
Add and subtract terms
E(X)=i=1∑nxi⋅P(xi)
The expected value of a random variable does not necessarily have to be equal to a possible value of the random variable. The alternative notations for the expected value are shown.
The table shows the random variable X assigned to outcomes of a probability experiment and the corresponding probabilities. Calculate the expected value of the random variable. If necessary, round the answer to two decimal places.
Jordan will take a multiple-choice test in which each question has five choices. She receives 1 point for each correct answer and loses 0.5 points for an incorrect answer. Help Jordan decide whether guessing is advantageous or not for the questions she does not know the correct answer.
No, it is not advantageous to guess. See solution.
Define a random variable for the situation. Then, find the expected value of it.
X | 1 | -0.5 |
---|---|---|
P(X) | 51 | 54 |
Substitute values
Tearrik and six more people apply for a job. The company announces that four of the applicants will be hired using random selection. It is known that four out of the seven applicants are females.
Tearrik hears that the company filled the four opening with all four female applicants. Tearrik wonders if this is an unlikely outcome. By answering the following questions, help Tearrik understand if the outcome, indeed, was unlikely.
n=7, r=4
Subtract term
Write as a product
Cancel out common factors
Simplify quotient
Write as a product
Multiply
Calculate quotient
Rewrite 4C1 as 1!(4−1)!4!
Rewrite 3C3 as 3!(3−3)!3!
Subtract term
Multiply fractions
Write as a product
Cancel out common factors
Simplify quotient
1!=1
0!=1
a⋅1=a
1a=a
Number of Possible Outcomes, 35 | |||
---|---|---|---|
1 Female and 3 Males |
2 Females and 2 Males |
3 Females and 1 Male |
4 Females and 0 Males |
4C1⋅3C3 | 4C2⋅3C2 | 4C3⋅3C1 | 4C4⋅3C0 |
4⋅1 | 6⋅3 | 4⋅3 | 1⋅1 |
4 | 18 | 12 | 1 |
Considering the above table, the probabilities can be determined.
X | 1 | 2 | 3 | 4 |
---|---|---|---|---|
P(X) | 354 | 3518 | 3512 | 351 |
Substitute values
a⋅cb=ca⋅b
Add fractions
Calculate quotient
Round to nearest integer
The expected values of situations can be compared when making decisions. Now, returning to the initial challenge, the most profitable cryptocurrency for Kevin to invest in can be determined. Recall the table showing the probabilities of the events over the next week.
Lose $25 | Gain $10 | Gain $50 | |
---|---|---|---|
Coin Q | 0.40 | 0.20 | 0.40 |
Coin R | 0.10 | 0.75 | 0.15 |
Start by calculating the expected value of the gain or loss for each cryptocurrency.
The expected value of gain or loss for each cryptocurrency will be calculated one at a time.
Let X be the random variable that represents the gains and losses for Coins Q.
X | -25 | 10 | 50 |
---|---|---|---|
P(X) | 0.4 | 0.2 | 0.4 |
Substitute values
Let Y be the random variable that represents the gains and losses for Coin R.