We now demonstrate the above results with a tree diagram.
Suppose a jar contains 3 red and 4 white marbles. If two marbles are drawn without replacement, find the following probabilities using a tree diagram.
The probability that both marbles are white.
The probability that the first marble is red and the second white.
The probability that one marble is red and the other white.
Let
$R$ be the event that the marble drawn is red, and let
$W$ be the event that the marble drawn is white.
We draw the following tree diagram.
Although the tree diagrams give us better insight into a problem, they are not practical for problems where more than two or three things are chosen. In such cases, we use the concept of combinations that we learned in
[link] . This method is best suited for problems where the order in which the objects are chosen is not important, and the objects are chosen without replacement.
Suppose a jar contains 3 red, 2 white, and 3 blue marbles. If three marbles are drawn without replacement, find the following probabilities.
$P\left(\text{Two red and one white}\right)$
$P\left(\text{One of each color}\right)$
$P\left(\text{None blue}\right)$
$P\left(\text{At least one blue}\right)$
Let us suppose the marbles are labeled as
${R}_{1}$ ,
${R}_{2}$ ,
${R}_{3}$ ,
${W}_{1}$ ,
${W}_{2}$ ,
${B}_{1}$ ,
${B}_{2}$ ,
${B}_{3}$ .
$P\left(\text{Two red and one white}\right)$
We analyze the problem in the following manner.
Since we are choosing 3 marbles from a total of 8, there are
$\mathrm{8C3}=\text{56}$ possible combinations. Of these 56 combinations, there are
$\mathrm{3C2}\times \mathrm{2C1}=6$ combinations consisting of 2 red and one white. Therefore,
$P\left(\text{Two red and one white}\right)=\frac{\mathrm{3C2}\times \mathrm{2C1}}{\mathrm{8C3}}=\frac{6}{\text{56}}$ .
$P\left(\text{One of each color}\right)$
Again, there are
$\mathrm{8C3}=\text{56}$ possible combinations. Of these 56 combinations, there are
$\mathrm{3C1}\times \mathrm{2C1}\times \mathrm{3C1}=\text{18}$ combinations consisting of one red, one white, and one blue. Therefore,
$P\left(\text{One of each color}\right)=\frac{\mathrm{3C1}\times \mathrm{2C1}\times \mathrm{3C1}}{\mathrm{8C3}}=\frac{\text{18}}{\text{56}}$ .
By "at least one blue marble," we mean the following: one blue marble and two non-blue marbles, or two blue marbles and one non-blue marble, or all three blue marbles. So we have to find the sum of the probabilities of all three cases.
$P\left(\text{At least one blue}\right)=P\left(\text{one blue, two non-blue}\right)+P\left(\text{two blue, one non-blue}\right)+P\left(\text{three blue}\right)$
$P\left(\text{At least one blue}\right)=\frac{\mathrm{3C1}\times \mathrm{5C2}}{\mathrm{8C3}}+\frac{\mathrm{3C2}\times \mathrm{5C1}}{\mathrm{8C3}}+\frac{\mathrm{3C3}}{\mathrm{8C3}}$
$P\left(\text{At least one blue}\right)=\text{30}/\text{56}+\text{15}/\text{56}+1/\text{56}=\text{46}/\text{56}=\text{23}/\text{28}$ .
Alternately,
we use the fact that
$P\left(E\right)=1-P\left({E}^{c}\right)$ .
If the event
$E=\text{At least one blue}$ , then
${E}^{c}=\text{None blue}$ .
But from part c of this example, we have
$\left({E}^{c}\right)=5/\text{28}$
Five cards are drawn from a deck. Find the probability of obtaining two pairs, that is, two cards of one value, two of another value, and one other card.
Let us first do an easier problem–the probability of obtaining a pair of kings and queens.
Since there are four kings, and four queens in the deck, the probability of obtaining two kings, two queens and one other card is
$P\left(\text{A pair of kings and queens}\right)=\frac{\mathrm{4C2}\times \mathrm{4C2}\times \text{44}\mathrm{C1}}{\text{52}\mathrm{C5}}$
To find the probability of obtaining two pairs, we have to consider all possible pairs.
Since there are altogether 13 values, that is, aces, deuces, and so on, there are
$\text{13}\mathrm{C2}$ different combinations of pairs.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
8. It is known that 80% of the people wear seat belts, and 5% of the people quit smoking last year. If 4% of the people who wear seat belts quit smoking, are the events, wearing a seat belt and quitting smoking, independent?
Mr. Shamir employs two part-time typists, Inna and Jim for his typing needs. Inna charges $10 an hour and can type 6 pages an hour, while Jim charges $12 an hour and can type 8 pages per hour. Each typist must be employed at least 8 hours per week to keep them on the payroll. If Mr. Shamir has at least 208 pages to be typed, how many hours per week should he employ each student to minimize his typing costs, and what will be the total cost?
At De Anza College, 20% of the students take Finite Mathematics, 30% take Statistics and 10% take both. What percentage of the students take Finite Mathematics or Statistics?