The probabilities that each of two independent experiments will have a successful outcome are \(\frac{8}{15}\) and \(\frac{2}{3}\), respectively. What is the probability that both experiments will have successful outcomes?
Show Hint
- \(\frac{4}{5}\)
- \(\frac{6}{5}\)
- \(\frac{2}{15}\)
- \(\frac{16}{45}\)
- \(\frac{64}{225}\)
The Correct Option is D
Solution and Explanation
This question deals with the probability of independent events. Two events are independent if the outcome of one does not affect the outcome of the other. To find the probability that both independent events occur, we multiply their individual probabilities.
Step 2: Key Formula or Approach:
If \(A\) and \(B\) are two independent events, the probability that both \(A\) and \(B\) occur is given by the formula:
\[ P(A \text{ and } B) = P(A) \times P(B) \] Step 3: Detailed Explanation:
Let \(P(S_1)\) be the probability of success for the first experiment and \(P(S_2)\) be the probability of success for the second experiment.
We are given:
\[ P(S_1) = \frac{8}{15} \] \[ P(S_2) = \frac{2}{3} \] We need to find the probability that both experiments are successful. Since the experiments are independent, we multiply their probabilities:
\[ P(S_1 \text{ and } S_2) = P(S_1) \times P(S_2) = \frac{8}{15} \times \frac{2}{3} \] Now, we multiply the numerators together and the denominators together:
\[ \frac{8 \times 2}{15 \times 3} = \frac{16}{45} \] Step 4: Final Answer:
The probability that both experiments will have successful outcomes is \(\frac{16}{45}\).
Top Questions on Probability
- Grace has 16 jellybeans in her pocket. She has 8 red ones, 4 green ones, and 4 blue ones. What is the minimum number of jellybeans she must take out of her pocket to ensure that she has one of each color?
- Master Chef Alan makes a dish every day from one of his recipe books. He has written 3 books and each book contains 15 different recipes. What is the probability that he will cook 4th dish from 3rd book today?
- At a medical research lab, nine doctors are conducting multiple clinical trials. Six of the doctors are working on a clinical trial with exactly one other doctor and three doctors are working on a clinical trial with exactly two other doctors. If two doctors are selected at random from the lab, what is the probability that those two doctors are NOT working together on a clinical trial?
- Defects on plywood sheet occur at random with the average of one defect per 50 sq.ft. Find the probability that such a sheet has : (a) no defect (b) at least one defect (use \(e^{-1} = 0.3678\))
- Find the mean of number of heads in three tosses of a fair coin.
Questions Asked in GRE exam
In early-twentieth-century England, it was fashionable to claim that only a completely new style of writing could address a world undergoing unprecedented transformation — just as one literary critic recently claimed that only the new ”aesthetic of exploratory excess” can address a world undergoing . . . well, you know. Yet in early-twentieth-century England, T. S. Eliot, a man fascinated by the ”presence” of the past, wrote the most innovative poetry of his time. The lesson for today’s literary community seems obvious: a reorientation toward tradition would benefit writers no less than readers. But if our writers and critics indeed respect the novel’s rich tradition (as they claim to), then why do they disdain the urge to tell an exciting story?
- GRE - 2024
- Reading Comprehension
In the 1970’s, two debates engaged many scholars of early United States history. One focused on the status of women, primarily White women. Turning on the so-called golden age the ory, which posited that during the eighteenth-century colonial era, American women enjoyed a brief period of high status relative to their English contemporaries and to nineteenth-century American women, this debate pitted scholars who believed women’s lives deteriorated after 1800 against those who thought women’s lives had been no better before 1800. At issue were the causes of women’s subordination: were these causes already in place when the English first settled North America or did they emerge with the rise of nineteenth-century industrial capital ism? The second debate, the so-called origins debate, concerned the emergence of racial slavery in the southern colonies: was slavery the inevitable result of the deep-rooted racial prejudice of early British colonists or did racial prejudice arise only after these planters instituted slave labor?
Although these debates are parallel in some respects, key differences distinguished them. Whereas the debate over women’s status revolved around implicit comparisons of colonial women to their counterparts in the antebellum period (1800-1860), thus inviting comment from scholars of both historical periods, the origins debate was primarily confined to a discussion about slavery in colonial America. Second, in contrast to the newness of the debate over women’s status and its continued currency throughout the early 1980’s, the debate over race and slavery, begun in the 1950’s, had lost some of its urgency with the publication of Morgan’s American Slavery, American Freedom (1975), widely regarded as the last word on the subject. Each debate also assumed a different relationship to the groups whose histories it concerned. In its heyday, the origins debate focused mainly on White attitudes toward Africans rather than on Africans themselves. With few exceptions, such as Wood’s Black Majority (1974) and Mullin’s Flight and Rebellion (1972), which were centrally concerned with enslaved African men, most works pertaining to the origins debate focused on the White architects, mostly male, of racial slavery. In contrast, although women’s historians were interested in the institutions and ide ologies contributing to women’s subordination, they were equally concerned with documenting women’s experiences. As in the origins debate, however, early scholarship on colonial women defined its historical constituency narrowly, women’s historians focusing mainly on affluent White women.
Over time, however, some initial differences between the approaches taken by scholars in the two fields faded. In the 1980’s, historians of race and slavery in colonial America shifted their attention to enslaved people; interest in African American culture grew, thereby bringing en slaved women more prominently into view. Historians of early American women moved in similar directions during the decade and began to consider the effect of racial difference on women’s experience.- GRE - 2024
- Reading Comprehension
The most plausible justification for higher taxes on automobile fuel is that fuel consumption harms the environment and thus adds to the costs of traffic congestion. But the fact that burning fuel creates these ”negative externalities” does not imply that no tax on fuel could ever be too high. Economics is precise about the tax that should, in principle, be levied to deal with negative externalities: the tax on a liter of fuel should be equal to the harm caused by using a liter of fuel. If the tax is more than that, its costs (including the inconvenience to those who would rather have used their cars) will exceed its benefits (including any reduction in congestion and pollution).
- GRE - 2024
- Reading Comprehension
Objectively, of course, the various ecosystems that sustain life on the planet proceed independently of human agency, just as they operated before the hectic ascendancy of Homo sapiens. But it is also true that it is difficult to think of a single such system that has not, for better or worse, been substantially modified by human culture. Nor is this simply the work of the industrial centuries. It has been happening since the days of ancient Mesopotamia. It is coeval with the origins of writing, and has occurred throughout our social existence. And it is this irreversibly modified world, from the polar caps to the equatorial forests, that is all the nature we have.
- GRE - 2024
- Reading Comprehension
Music critics have consistently defined James P. Johnson as a great early jazz pianist, originator of the 1920’s Harlem ”stride” style, and an important blues and jazz composer. In addition, however, Johnson was an innovator in classical music, composing symphonic music that incor porated American, and especially African American, traditions.
Such a blend of musical elements was not entirely new: by 1924 both Milhaud and Gershwin had composed classical works that incorporated elements of jazz. Johnson, a serious musician more experienced than most classical composers with jazz, blues, spirituals, and popular music, was particularly suited to expand Milhaud’s and Gershwin’s experiments. In 1927 he completed his first large-scale work, the blues- and jazz-inspired Yamekraw, which included borrowings from spirituals and Johnson’s own popular songs. Yamekraw, premiered successfully in Carnegie Hall, was a major achievement for Johnson, becoming his most frequently performed extended work. It demonstrated vividly the possibility of assimilating contemporary popular music into the symphonic tradition.- GRE - 2024
- Reading Comprehension