Joanna Benson

2021-12-20

Seven balls are randomly withdrawn from an urn that contains 12 red, 16 blue, and 18 green balls. Find the probability that (a) 3 red, 2 blue, and 2 green balls are withdrawn; (b) at least 2 red balls are withdrawn; (c) all withdrawn balls are the same color; (d) either exactly 3 red balls or exactly 3 blue balls are withdrawn.

Barbara Meeker

Beginner2021-12-21Added 38 answers

Step 1

Given:

An urn that contains 12R, 16B, 18G balls

7 balls are randomly chosen

The result space S exhibits an equal distribution of probabilities.

$S=\{\{{x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5},{x}_{6},{x}_{7}\};{x}_{i}$ are different balls from the urn

S having 7 different valued combinations $\left|S\right|=$

$(\genfrac{}{}{0ex}{}{46}{7})$

($\left|X\right|$ is the number of elements in X)

For an event $A\subseteq S$, $\left|A\right|=n$

$P(A)=\frac{n}{(\genfrac{}{}{0ex}{}{46}{7})}$

a) $P(3R,2B,2G)=?$

Find the number of ball combinations with 7 different values where 3 are red, 2 are blue, and 2 are green.

Choose the three red (R) from 12 of them in $(\genfrac{}{}{0ex}{}{12}{3})$ ways, then choose 2 more blue to generate $(\genfrac{}{}{0ex}{}{16}{2})$ times more possibilities, and choose the 2 green ones - $(\genfrac{}{}{0ex}{}{18}{2})$ possibilities

$P(3R,2B,2G)=\frac{(\genfrac{}{}{0ex}{}{12}{3})\cdot (\genfrac{}{}{0ex}{}{16}{2})\cdot (\genfrac{}{}{0ex}{}{18}{2})}{(\genfrac{}{}{0ex}{}{46}{7})}=\frac{3060}{40549}=0.075464$

zesponderyd

Beginner2021-12-22Added 41 answers

Step 2

b) $$P(At least two R)=?

Remember Proposition 4.1

$P\left(A\right)=1-P\left({A}^{c}\right)$

A draw of 0 or 1 red ball is a complement to an event in which at least 2 red balls were drawn.

$P(0\text{}R\text{}\mathit{\text{or}}\text{}R)=P(0R)+P(1R)\Rightarrow $ mutually exclusive events

There are $(\genfrac{}{}{0ex}{}{34}{7})$a selection of 7 balls without any reds (that is choosing between 34 non red balls)

And if there is only one red ball, it can be chosen from the red balls in 12 different ways, but for every red ball selection, the other six balls can be selected in $(\genfrac{}{}{0ex}{}{34}{6})$ different ways.

$$\begin{array}{cc}P(At\text{}least\text{}two\text{}R)& =1-P(0Ror1R)\\ & =1-P(0R)-P(1R)\\ & =1-\frac{(\genfrac{}{}{0ex}{}{34}{7})}{(\genfrac{}{}{0ex}{}{46}{7})}-\frac{12(\genfrac{}{}{0ex}{}{34}{6})}{(\genfrac{}{}{0ex}{}{46}{7})}\\ & =0.59797...\end{array}$$

nick1337

Expert2021-12-27Added 777 answers

Step 3

c) PP(7R or 7B or 7G)=?

P(7R or 7B or 7G)= P(7R)+P(7B)+P(7G) $\Rightarrow $ mutually exclusive events

The reds can be chosen in $(\genfrac{}{}{0ex}{}{12}{7})$ ways because there are 12 red balls to chose from, similarly the blue balls can be any of the $(\genfrac{}{}{0ex}{}{16}{7})$ possible choices, and 7G has $(\genfrac{}{}{0ex}{}{18}{7})$ possibilities.

Step 4

d) P(precisely 3 R or precisely 3 B)=?

A - event that excatly 3 red balls are drawn

B - event that excatly 3 blue balls are drawn

The sum of the selections for the three red balls produces the number of options where - $(\genfrac{}{}{0ex}{}{12}{3})$ and the number of choices for the remaining non red balls - $(\genfrac{}{}{0ex}{}{34}{4})$

Same for the blue, the number of draws of precisely three blue balls is $(\genfrac{}{}{0ex}{}{16}{3})(\genfrac{}{}{0ex}{}{30}{4})$

Additionally, if there are three red balls and three blue balls among the seven balls available $(A\cap BA\cap B)$ it can be any of $(\genfrac{}{}{0ex}{}{12}{3})(\genfrac{}{}{0ex}{}{16}{3})(\genfrac{}{}{0ex}{}{18}{1})$

Due to the fact that the initial calculation $A\cup B$ is the Proposition 4.4.

$\begin{array}{cc}P(precisely\text{}3\text{}R\text{}or\text{}precisely\text{}3\text{}B)& =P(A\cup B)\\ & =P(A)+P(B)-P(A\cap B)\\ & =\frac{(\genfrac{}{}{0ex}{}{12}{3})(\genfrac{}{}{0ex}{}{34}{4})}{(\genfrac{}{}{0ex}{}{46}{7})}+\frac{(\genfrac{}{}{0ex}{}{16}{3})(\genfrac{}{}{0ex}{}{30}{4})}{(\genfrac{}{}{0ex}{}{46}{7})}-\frac{(\genfrac{}{}{0ex}{}{12}{3})(\genfrac{}{}{0ex}{}{16}{3})18}{(\genfrac{}{}{0ex}{}{46}{7})}\\ & =0.4359...\end{array}$

Assume that when adults with smartphones are randomly selected, 54% use them in meetings or classes (based on data from an LG Smartphone survey). If 8 adult smartphone users are randomly selected, find the probability that exactly 6 of them use their smartphones in meetings or classes?

Write formula for the sequence of -4, 0, 8, 20, 36, 56, 80, where the order of f(x) is 0, 1, 2, 3, 4, 5, 6 respectively

A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment.n=20,

p=0.7,

x=19

P(19)=

In binomial probability distribution, the dependents of standard deviations must includes.

a) all of above.

b) probability of q.

c) probability of p.

d) trials.The probability that a man will be alive in 25 years is 3/5, and the probability that his wifewill be alive in 25 years is 2/3

Determine the probability that both will be aliveHow many different ways can you make change for a quarter??

(Different arrangements of the same coins are not counted separately.)One hundred people line up to board an airplane that can accommodate 100 passengers. Each has a boarding pass with assigned seat. However, the first passenger to board has misplaced his boarding pass and is assigned a seat at random. After that, each person takes the assigned seat. What is the probability that the last person to board gets his assigned seat unoccupied?

A) 1

B) 0.33

C) 0.6

D) 0.5The value of $(243{)}^{-\frac{2}{5}}$ is _______.

A)9

B)$\frac{1}{9}$

C)$\frac{1}{3}$

D)01 octopus has 8 legs. How many legs does 3 octopuses have?

A) 16

B 24

C) 32

D) 14From a pack of 52 cards, two cards are drawn in succession one by one without replacement. The probability that both are aces is...

A pack of cards contains $4$ aces, $4$ kings, $4$ queens and $4$ jacks. Two cards are drawn at random. The probability that at least one of these is an ace is A$\frac{9}{20}$ B$\frac{3}{16}$ C$\frac{1}{6}$ D$\frac{1}{9}$

You spin a spinner that has 8 equal-sized sections numbered 1 to 8. Find the theoretical probability of landing on the given section(s) of the spinner. (i) section 1 (ii) odd-numbered section (iii) a section whose number is a power of 2. [4 MARKS]

If A and B are two independent events such that $P(A)>0.5,P(B)>0.5,P(A\cap \overline{B})=\frac{3}{25}P(\overline{A}\cap B)=\frac{8}{25}$, then the value of $P(A\cap B)$ is

A) $\frac{12}{25}$

B) $\frac{14}{25}$

C) $\frac{18}{25}$

D) $\frac{24}{25}$The unit of plane angle is radian, hence its dimensions are

A) $[{M}^{0}{L}^{0}{T}^{0}]$

B) $[{M}^{1}{L}^{-1}{T}^{0}]$

C) $[{M}^{0}{L}^{1}{T}^{-1}]$

D) $[{M}^{1}{L}^{0}{T}^{-1}]$Clinical trial tests a method designed to increase the probability of conceiving a girl. In the study, 340 babies were born, and 289 of them were girls. Use the sample data to construct a 99% confidence interval estimate of the percentage of girls born?