A: Between 1200 and 2400, this will happen for 11 times. So divide 12 hr by 11 and add to 1200, i.e. 13:05:27
Q: A cell can do 4 things: 1) Dies; 2) Nothing; 3) Duplicate; or 4) Triplicate. What is P(colony dies out) if we start with one single cell?
A: Solve the polynomial
$ p = \frac {1} {4} + \frac {1}{4} p + \frac {1}{4} p^2 + \frac {1}{4} p^3$
Q: A 6x6 matrix has entries of either +1 or -1. How many unique combinations are there such that all column products and row products are +1?
A: Consider the top left 5x5 sub-matrix. It can take any permutation (as long as each row/column contains no more than 3 same number) because we can rely on the rest of the 6x6 matrix (an "L" shape) to "fix" it. So the answer is $ 2^{25} $
Q: There is an integer X. The product of the digits is 96. What is the largest/smallest possible value of X?
A: smallest = 268, largest = 322222
Q: What is the sum of all digits of all the numbers between 1 and 1000000?
A: Convince yourself that all digits appear for an equal number of times. The answer is 27000001
Q: A bus makes a stop, and 3/4 of all passengers get off, while 7 get on. This repeats for two more times. What is the smallest number of initial passengers?
A: 52
Q: Paint the outside of a cube, then dice it into 27 equal smaller cubes. Pick one at random and roll it. What is P(no paint visible after the roll)?
A: $ 6/27 \times 1/6 + 1/27 $
Q: Two equally talented teams are playing a series of games in which the first team to win 4 games wins the series. What is P(7th game has to be played)?
A: $ \frac {6!}{3!3!} \frac {1}{2^6} = \frac {5}{16}$
Q: I roll a die, spin a roulette and draw a card from a deck of 52. What is the probability that all three give the same number?
A: $ 5 \times \frac 1 6 \times \frac 1 {13} \times 1 {38} $
Q: What is the last digit of 3 to the power 33?
A: Some experimentation would indicate that $3^0=1$, $3^1=3$, $
Q: N couples shake hand with others. It is known that 1) They don't shake hand with their own partners; and 2) The first (2N - 1) people surveyed all have different number of handshakes. What is the number of handshake the remaining person made?
A: N - 1 (Can draw graphs to deduce; any analytic solution?)
Q: What is P(3 random points on a circle are within an 180 degree arc)?
A: There are two approaches: 1) Area on an abstract x-y plane, x and y are distances from the first point pinned down (call it z, which is arbitrary); or 2) $ N \times \frac {1}{2^{N-1}} $, with $ N=3$. Both give the answer of 3/4.
Q: How do you simulate a 6-sided die using a fair coin? What's the expected number of toss to produce the number?
A: Use a binary scheme. With 3 tosses, there are 8 possible outcomes, which is more than enough to simulate 6 sides. The trick is to know which 2 to exclude. {HHH,TTT} would be a bad choice because you always have to wait until the 3rd toss to know if you failed. So we should choose something like {HHH,HHT} or {THH,THT}. The expected number of toss is given by solving
$E[N]=\frac 6 8 \times 3 + \frac 2 8 (2+E[N])$
Q: Place 3 points randomly on the perimeter of a square. What is P(they don't form a triangle) and P(all lie on different edges)?
A: $P(they_don't
__Q: 10 light bulbs are lined up in a row, and it is known that no two of the adjacent light bulbs can be both on. What is the possible number of arrangements?
A: Fibonacci sequence, 144
Q: You have a large number of red and blue balls (equally many) in a large container. What is P(odd number of red) if you draw a) 3 balls from the container? b) 10 balls from the container?
A: 0.5 regardless of how many you draw. For example if 3 are drawn, the binomial distribution is 1(odd) 3(even) 3(odd) 1(even); if 4 are drawn, the binomial distribution is 1(even) 4(odd) 6(even) 4(odd) 1(even).
Q: Transport as many apples as possible to a destination 1000 km away starting with 3000 apples. For each km of transportation, you lose 1 apple.
A: The best tactic is to always start with full load so as to maximize number of "apple km" traveled per apple lost. So the 1st stop is optimally at 3000 - 3X = 2000 <=> X = 333; the 2nd stop is optimally at 2000 - 2X = 1000 <=> X = 500. After 833 km, we are left with 1000 apples. Hence we will have 833 apples at the destination.
Q: You are playing against two players A and B individually in a tournament, with A being the better player. If you have to win at least 2 consecutive games out of 3 in order to be the champion, would you prefer the order ABA or BAB?
A: Out of the 8 outcomes {LLL,LLW,LWL,LWW,WLL,WLW,WWL,WWW}, only {WWW,WWL,LWW} are desirable. If q = P(winning against A) and p = P(winning against B) so that p > q,
P(champion with the order ABA) = pqp + pq(1-p) + (1-p)qp = 2pq - ppq
P(champion with the order BAB) = qpq + qp(1-q) + (1-q)pq = 2pq - pqq
So P(champion with the order BAB) > P(champion with the order ABA)
Hi,
ReplyDeleteI couldn't find a logical way to get this question
Q: What is the last digit of 3 to the power 33?
A: 3
So doing the actual calculation, the result is 5559060566555520 and does not end by a 3.
Where did you get the answer from and what is the rationale behind it ?
Cheers
Hi Tristan,
DeleteThanks for your comment, I have updated the post in response. Basically what happened was that the calculator you used suffered from overflow. If you do "3^33 mod 10" in Google search you'll find that the answer is in fact 3. By the way, 5559060566555520 is an even number and hence cannot possibly be the right answer.
Cheers!