MATHEMATICS COLLEGE

A particular concentration of a chemical found in polluted water has been found to be lethal to 40% of the crayfish that are exposed to the concentration for 24 hours. 22 crayfish are placed in a tank containing this concentration of chemical in the water. (a) What is the probability that 13 or 17 survive?
(b) What is the probability that at least 17 survive?
(c) What is the probability that at most 16 survive?
(d) What number of crayfish are expected to survive?
(e) What is the variance in the number of crayfish that are expected to survive? No actual crayfish were harmed in the making of this question.

Answers

Answer 1

Answer:

Answer:

a) $P(X=13)=(24C13)(0.4)^(13) (1-0.4)^(24-13)=0.0608$

$P(X=17)=(24C17)(0.4)^(17) (1-0.4)^(24-17)=0.0017$

$P(X=13 U X=17) = 0.0608+0.0017=0.0624$

b) $P(X \geq 17)=0.000427$

c) $P(X \geq 16)= 1-0.000427=0.9996$

d) $E(X)= np = 22*0.4=8.8$

e) $Var(X) = np(1-p)= 22*0.4*(1-0,4)=5.28$

Step-by-step explanation:

1) Previous concepts

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

2) Solution to the problem

Let X the random variable of interest, on this case we now that:

$X \sim Binom(n=24, p=0.4)$

The probability mass function for the Binomial distribution is given as:

$P(X)=(nCx)(p)^x (1-p)^(n-x)$

Where (nCx) means combinatory and it's given by this formula:

$nCx=(n!)/((n-x)! x!)$

3) Part a

$P(X=13)=(22C13)(0.4)^(13) (1-0.4)^(22-13)=0.0336$

$P(X=17)=(22C17)(0.4)^(17) (1-0.4)^(22-17)=0.00035$

$P(X=13 U X=17) = 0.0336+0.00035=0.0340$

4) Part b

$P(X \geq 17)=P(X=17)+P(X=18)+P(X=19)+P(X=20)+P(X=21)+P(X=22)$

$P(X=17)=(22C17)(0.4)^(17) (1-0.4)^(22-17)=0.00035$

$P(X=18)=(22C18)(0.4)^(18) (1-0.4)^(22-18)=0.0000651$

$P(X=19)=(22C19)(0.4)^(19) (1-0.4)^(22-19)=0.00000914$

$P(X=20)=(22C20)(0.4)^(20) (1-0.4)^(22-20)=0.000000914$

$P(X=21)=(22C21)(0.4)^(21) (1-0.4)^(22-21)=5.81x10^(-8)$

$P(X=22)=(22C22)(0.4)^(22) (1-0.4)^(22-22)=1.76x10^(-9)$

$P(X \geq 17)=0.000427$

5) Part c

$P(X \leq 16)=1-P(X>16)=1-P(X\geq 17) = 1-[P(X=17)+P(X=18)+P(X=19)+P(X=20)+P(X=21)+P(X=22)]$

$P(X \geq 16)= 1-0.000427=0.9996$

6) Part d

The expected value is:

$E(X)= np = 22*0.4=8.8$

7) Part e

The variance is given by:

$Var(X) = np(1-p)= 22*0.4*(1-0,4)=5.28$

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What is the sum in simplest form?
4 1/2 + 1 3/5

Answers

Answer:

23 1/10

Step-by-step explanation:

Final answer:

To find the sum in simplest form of 4 1/2 and 1 3/5, you convert each to an improper fraction, adding the fractions and convert back to a mixed number. The sum is 6 1/10.

Explanation:

The sum in simplest form of the two numbers 4 1/2 and 1 3/5 can be calculated as follows:

Change each mixed number into an improper fraction. 4 1/2 becomes 9/2 and 1 3/5 becomes 8/5.
Then add the two fractions: 9/2 + 8/5 equals 45/10 + 16/10 (after finding a common denominator), which equals 61/10.
Finally, convert back to a mixed number to get 6 1/10.

So, the sum in the simplest form of 4 1/2 and 1 3/5 is 6 1/10.

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Records show that the average number of phone calls received per day is 9.2. Find the probability of between 2 and 4 phone calls received (endpoints included) in a given day.

Answers

Answer:

4.76% probability of between 2 and 4 phone calls received (endpoints included) in a given day.

Step-by-step explanation:

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:

$P(X = x) = (e^(-\mu)*\mu^(x))/((x)!)$

In which

x is the number of sucesses

e = 2.71828 is the Euler number

$\mu$ is the mean in the given interval.

Records show that the average number of phone calls received per day is 9.2.

This means that $\mu = 9.2$ .

Find the probability of between 2 and 4 phone calls received (endpoints included) in a given day.

$(2 \leq X \leq 4) = P(X = 2) + P(X = 3) + P(X = 4)$

$P(X = x) = (e^(-\mu)*\mu^(x))/((x)!)$

$P(X = 2) = (e^(-9.2)*(9.2)^(2))/((2)!) = 0.0043$

$P(X = 3) = (e^(-9.2)*(9.2)^(3))/((3)!) = 0.0131$

$P(X = 4) = (e^(-9.2)*(9.2)^(4))/((4)!) = 0.0302$

$(2 \leq X \leq 4) = P(X = 2) + P(X = 3) + P(X = 4) = 0.0043 + 0.0131 + 0.0302 = 0.0476$

4.76% probability of between 2 and 4 phone calls received (endpoints included) in a given day.

Divide the fractions:

5/8 divided by 1/24 =

Answers

Answer:

The answer to this question is 15.

Step-by-step explanation:

We know that dividing a fraction by another fraction is the same as multiplying a fraction by the other fractions reciprocal, where the numerator and denominator are switched.

5/8 ÷ 1/24 =

5/8 x 24/1 =

120/8

120 ÷ 8 = 15.

Hence, the answer to this question is 15.

Hope this helps! :D

Final Answer:

Step-by-step explanation:

5/8 ÷ 1/24

5/8 ÷ 24/1

5/8 × 24/1 = 120/8

120/8 ⇒ 15/1 ⇒ 15

A state's license plates consist of four letters followed by three numerals, and 246 letter arrangements are not allowed. how many plates can the state issue?

Answers

The state will issue 4,56,97,354 plates

What is exponents and powers?

When anumber is multiplied by itself a finite number of times, the number being multiplied is called the base number, and the number of times it is being multiplied is called the exponent. Suppose a number is being multiplied by itself n number of times, then this expression is called the nth power of the given number.

An exponent is the number of times a given number is being multiplied by itself.
The expression when a number is being multiplied by itself n number of times is called the nth power of the given number.

Given:

Number of letters = 4

Number of numerals = 3

As 246 letter arrangements are not allowed.

So, we have total 26 letters.

and numbers from 0 to 9.

So, the number of plates will be

= 26*26*26*26*10*10

=(26)^4 * (10)^2

= 45697600

Now, deducted those 246 letter arrangements which are not allowed

= 45697600 - 246

= 4,56,97,354

Hence, the state will issue 4,56,97,354 plates.

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There are 26 letters in the alphabet, there are also 10 numbers 1-10. You multiply that and it would be 260.

If $11000 yields $1100 over 5 years, what is the interest rate?

Answers

Answer:

1.92449%

Step-by-step explanation:

Simple Interest Rate Formula: A = P(1 + r)ⁿ

Step 1: Define variables

Principle amount P = 11000

Total amount A = 11000 + 1100 = 12100

Years n = 5

Step 2: Substitute and Evaluate for r

12100 = 11000(1 + r)⁵

11/10 = (1 + r)⁵

1.01924 = 1 + r

r = 0.019245

Step 3: Convert to percentage

0.019245 × 100 = 1.92449%

I need help with this problem, please help!!!

Answers

Answer:

Yes. (See below for explanation.)

Step-by-step explanation:

The number of servings is found by dividing the quantity available by the size of a serving. The quantity of punch is the sum of the quantities of the juices that go into the punch. The serving size of 3/4 cup is the same as 6 ounces, since a cup is 8 ounces. (3/4 × 8 oz = 6 oz)

The quantity available is (64 oz + 28 oz + 76 oz). The serving size is 6 oz. Since the units of numerator and denominator are the same, they cancel, leaving ...

... number of servings = (quantity available)/(serving size)

... = (64 +28 +76)/6 . . . . as shown in the problem statement

_____

It might not be obvious that the above ratio gives the number of servings. However, if you look at the real units, you see how it happens.

$(oz)/(((oz)/(serving)))= oz(serving)/(oz)=(oz)/(oz)serving=serving$

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