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How do theoretical probability and experimental probability relate?
Take for example, flipping a coin. Theoretically, if I flip it, there is a 50% chance that I flip a head and a a 50% chance that I flip a tail. That would lead us to believe that out of 100 flips, there should theoretically be 50 heads and 50 tails. But if you actually try this out, this may not be the case. What you actually get, say 46 heads and 54 tails, is the experimental probability. Thus, experimental probability differs from theoretical probability by the actual results. Where theoretical probability cannot change, experimental probability can.
You roll a pair of dice until you get doubles What is the probability that you get doubles in three or fewer rolls?
! So first of all we need to calculate the probability of the base event, that is to get a pair when rolling 2 dice. This is quite simple and we can see it in 2 ways. a) We throw the first dice, regardless of what it comes up as, we have 1 in 6 chanses that the second dice matches it i.e probability = 1/6 b) There are 6*6 possible out comes when rolling two dice. Out of those we can get pair of 1s, pair of 2s , pair of 3s, pair of 4s, pair of 5s or pair of 6s. That is 6 pairs out of 36 total = 1/6 probability. Now about making a sequence of throws. The probability of us making the pair in the first throw is as the basic event = 1/6. If we make it we dont continue and have reached our goal! If we dont get a pair (proability 1-1/6 = 5/6) we continue and make our second throw. Again we have a 1/6 chanse, so in total that we make it in exactly the 2nd throws is 5/6 (miss)*1/6 (hit) = 5/36. If we dont make it , we continue again. Now our chance in making it in exactly the 3rd time is 5/6 (miss)*5/6 (miss again) *1/6 (hit) = 25/216. So the probability of making a pair in three or fewer rolls is the sum of the above, so 1/6 + 5/36 + 25/216 = 91/216 = 0.42 (roughly) Another way when you have established the probability of a single action to find how many repeats you need is to use the Binomial or Poisson distribution (look it up). Best Regards
What is the probability that 4 randomly selected people all have different birthdays?
Let us assume that there are exactly 365 days in a year and that birthdays are uniformly randomly distributed across those days. First, what is the probability that 2 randomly selected people have different birthdays? The second person's birthday can be any day except the first person's, so the probability is 364/365. What is the probability that 3 people will all have different birthdays? We already know that there is a 364/365 chance that the first two will have different birthdays. The third person must have a birthday that is different from the first two: the probability of this happening is 363/365. We need to multiply the probabilities since the events are independent; the answer for 3 people is thus 364/365 × 363/365. You should now be able to solve it for 4 people.
What is the probability of picking a jack or a black three in one pick from a deck of cards?
Well there are six possibilities, the four jacks and the two black threes. This gives us a fraction of 6/52 which is a percentage of 11.54%
A coin is tossed and then a dice is rolled Find the probability of getting a 5 on the dice given that the coin landed tails up?
1/6 because there are six sides to a dice and we already know the coin landed tails up so that takes that equation out of the picture. otherwise we would be left with figuring the probability of the coin and the dice, but since the question tells us that the coin landed tails up the answer will then be 1/6
