Find the probability of getting at least one 3 when 9 fair dice are thrown
|
"It is the case that at least one is a three" is equivalent to saying "It is not the case that none are a three" Show
Here, we use the axioms of probability that $Pr(A\cup B) = Pr(A)+Pr(B)$ when $A$ and $B$ are mutually exclusive events, that $A\cup A^c = \Omega$ the sample space, and that $Pr(\Omega)=1$ to conclude that $Pr(\text{At least 1 three}) = 1-Pr(\text{no threes})$ If you wanted to, you could calculate it directly as follows: Break into cases based on the number of threes. "At least one three" is in this case equivalent to "Exactly one three or exactly two threes or exactly three threes or exactly four threes" Each of these are disjoint events, so $Pr(\text{at least one three}) = Pr(\text{exactly one three})+Pr(\text{exactly two threes})+Pr(\text{exactly three threes})+Pr(\text{exactly four threes})$ We calculate each individually. In the case of exactly one three, pick which location the three occupies (4 choices) and pick the number for each other location (5 choices each). Divide by the sample space size. Alternatively, approach via binomial distribution. Either way, you will arrive at $Pr(\text{exactly one three})=4(\frac{1}{6})(\frac{5}{6})^3$ Similarly, we have the others as $\binom{4}{k}(\frac{1}{6})^k(\frac{5}{6})^{4-k}$ for the probability of getting exactly $k$ threes. Adding, you have $Pr(\text{at least one three}) = \frac{4\cdot 5^3 + 6\cdot 5^2+4\cdot 5^1+4}{6^4}$, which after calculation you will find equals $1-(\frac{5}{6})^4$ The mistake of using $(\frac{1}{6})^4$ is that you calculated $Pr(\text{exactly four threes})$ which is only a small portion of the event at least one three. If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. François Ozon has remade a Fassbinder masterpiece, but the mystique and intrigue of the original is missing By Tim Robey The BBC’s only response to charges of systemic bias? Suck it upTwo damning reports have found evidence of what Roger Scruton called 'oikophobia' at the BBC. Yet the Corporation continues to play dumb Robin Aitken Why don’t we value our great cathedrals as museums?Our cathedrals are treasure-houses of beautiful art and artefacts, from the largest medieval world map to a millennium-old sundial Simon Heffer  What happened to feel-good films? Hollywood, here’s a New Year’s resolutionThis week, Victoria has been watching festive movies Victoria Coren Mitchell My 10 best films of 2022 – and my worstThis year gave us an unassailable argument for cinema as mass art – and The Nan Movie Robbie Collin The BBC’s only response to charges of systemic bias? Suck it upTwo damning reports have found evidence of what Roger Scruton called 'oikophobia' at the BBC. Yet the Corporation continues to play dumb Robin Aitken The first ordinary woman in English literature – how Chaucer’s Wife of Bath started a revolution
5/5 Neither a beautiful victim nor a hideous stereotype, the gap-toothed Wife of Bath is a woman for our times, says Marion Turner’s biography |
