Math and Arithmetic

Probability

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It is neither. If you repeated sets of 8 tosses and compared the number of times you got 6 heads as opposed to other outcomes, it would comprise proper experimental probability.

Dependent probability is the probability of an event which changes according to the outcome of some other event.

The probability of one event or the other occurring is the probability of one plus the probability of the other. The probability of getting 3 heads is the probability of 3 heads (1/23) multiplied by the probability of 4 tails (1/24) multiplied by the number of possible ways this could happen. This is 7c3 or 35. Thus the probability of 3 heads is 0.2734375. The probability of 2 tails is the probability of 2 tails (1/22) multiplied by the probability of 5 heads (1/25) multiplied by the number of ways this could happen. That is 7c5 or 21. Thus the probability of 2 tails is 0.1640625 The probability of one or the other is the sum of their probabilities: 0.1640625 + 0.2734375 = 0.4375 Thus the probability of getting 3 heads or 2 tails is 0.4375.

The probability to get heads once is 1/2 as the coin is fair The probability to get heads twice is 1/2x1/2 The probability to get heads three times is 1/2x1/2x1/2 The probability to get tails once is 1/2 The probability to get tails 5 times is (1/2)5 So the probability to get 3 heads when the coin is tossed 8 times is (1/2)3(1/2)5=(1/2)8 = 1/256 If you read carefully you'll understand that 3 heads and 5 tails has the same probability than any other outcome = 1/256 As the coin is fair, each side has the same probability to appear So the probability to get 3 heads and 5 tails is the same as getting for instance 8 heads or 8 tails or 1 tails and 7 heads, and so on

This is easiest calculated by calculating the probability that NO SINGLE heads is obtained; this is of course the complement of the question. The probability of this is 1/2 x 1/2 x 1/2 ... 7 times, in other words, (1/2)7. The complement, the probability that at least one head is obtained, is then of course 1 - (1/2)7, or a bit over 99%.

Your question is slightly vague, so I will pose a more defined question: What is the probability of 3 coin tosses resulting in heads exactly twice? This is a pretty easy question to answer. The three possible (winning) outcomes are: 1. Heads, Heads, Tails. 2. Heads, Tails, Heads. 3. Tails, Heads, Heads. If we look at the possible combination of other (losing) outcomes, we can easily determine the probability: 4. Heads, Heads, Heads. 5. Tails, Tails, Heads. 6. Tails, Heads, Tails. 7. Heads, Tails, Tails. 8. Tails, Tails, Tails. This means that to throw heads twice in 3 flips, we have a 3 in 8 chance. This is because there are 3 winning possibilities out of a total of 8 winning and losing possibilities.

999,999/2,000,000 for heads, and the same for tails * * * * * The above answer implies that there is a probability of 1/1,000,000 that the coin shows neither heads nor tails. It either stands on its rim or another image appears or the disappears into some other dimension or there is some other outcome. Not impossible, perhaps, but the probability of such an event is likely to be less than 1 in a million. So for all intents and purposes, if the coin is fair, Pr(H) = Pr(T) = 0.5

If two events are disjoint, they cannot occur at the same time. For example, if you flip a coin, you cannot get heads AND tails. Since A and B are disjoint, P(A and B) = 0 If A and B were independent, then P(A and B) = 0.4*0.5=0.2. For example, the chances you throw a dice and it lands on 1 AND the chances you flip a coin and it land on heads. These events are independent...the outcome of one event does not affect the outcome of the other.

1.525% in other words, NOT LIKELY

These are all independent events (flipping a coin will not affect the probability of drawing a Jack) so you can get the probability of all events occurring by multiplying together the probabilities of each event occurring. In other words: P (4 or 6, 2 heads, Jack) = P(4 or 6) * P(2 Heads) * P(Jack) Now we need to look at each probability separately. Remember that: Probability = Successful Outcomes / (Successful Outcomes + Unsuccessful Outcomes) In the case of rolling a die, a successful outcome (as defined in the problem) is rolling a 4 or 6. An unsuccessful outcome is everything else (1, 2, 3, or 5). Using the formula above then: Probability (4 or 6) = 2/6 = .33 Figuring out the probability of rolling two heads is slightly different because we are talking about two flips not one. In this case we have to go back to our original formula for multiple events. Probability (2 Heads) = Prob(Head) * Prob(Head) Since we know a coin-toss has a 1/2 chance of being heads or tails: Probability (2 Heads) = .5 * .5 = .25 Finally, in the case of picking up a card from a deck, a successful outcome (as defined in the problem) is picking a Jack. There are 4 Jacks in a standard deck so there are 4 possibilities of a successful outcome. There are 48 cards in a stardard deck that are not Jacks. Therefore: Probability (Jack) = 4/52 = .077 Now we can plug these values into our combination formula to get our answer. P (4 or 6, 2 heads, Jack) = P(4 or 6) * P(2 Heads) * P(Jack) P (4 or 6, 2 heads, Jack) = .33 * .25 * .077 = .00635 There is a .635% chance of rolling a 4 or 6, flipping a heads twice, AND drawing a Jack.

1/3 : One third, is the probability of 33.33% in other words 33% if your round. Hope I helped.

Two events are independent if the outcome of one has no effect on the probability of the outcomes for the other.

It's difficult to think of a real event to which an exact probability can be assigned. We say that flipping a coin yields 'heads' with probability 1/2 but we do not know that definitely. The only way of assigning a probability in the sense of numbers of heads versus total numbers of flips is by experiment. (Be aware though that there are other interpretations of the word probability.) If I were to flip a coin 500 times and obtained 249 heads then the experimental probability of obtaining a head would be 249/500 or 0.498.

It is 100%. The coin will result in heads or tails since there are no other possible outcomes.

States that to determine a probability, we multiply the probability of one event by the probability of the other event. Ex: Probability that two coins will land face heads up is 1/2 x 1/2 = 1/4 .

Two events are said to be independent if the outcome of one event does not affect the outcome of the other. Their probabilities are independent probabilities. If the events are not independent then they are dependent.

Probability is the likelyhood that some particular outcome will occur. It is expressed as a number between 0 and 1, with 0 meaning there is no likelyhood at all, and with 1 meaning it is guaranteed to happen.A simple example is the flipping of a coin. There is a 0.5 probability of getting heads, and there is a 0.5 probability of getting tails. Assuming a fair coin.Another example is a standard deck of 52 playing cards. There is a 0.25 probability of getting a Heart, and there is a 0.75 probability of not getting a Heart.Probability is not an exact predictor of outcome. If you flipped a coin 100 times, for instance, you would expect about 50 of the throws to be heads, but you will not get that exact outcome. Even if you throw the coin 10 million times, you will not get exactly 5 million heads. What happens is that, as the number of trials increases, the closer the experimental results are to the theoretical probability.One important factor of probability is that the sum of the probabilities of all possible outcomes is equal to the number of all possible trials. Stated another way, the sum of all probabilities is always equal to exactly 1. Stated yet another way, the probability that something is going to happen is 1.This can get complicated, particularly when the number of outcomes is large. Take the New York State lottery game Lotto, for instance - you pick 6 numbers out of 59 - there are so many popssible results that special mathematics (the science of permutations and combinations) is used to do that calculation. In this case, there are 45,057,474 combinations of 59 things taken 6 at a time, so the odds of winning the jackpot on one game is 1 is 45,057,474 or about a probability of 0.00000002194. Another way to say this is that the probability of not winning the jackpot is about 0.9999999778.I have only scratched the surface on this topic. Any other contributor is, as always, welcome to refine this answer.

A biased probability is one where not every outcome has the same chance of occurring. A biased coin is one where one side, the "heads" or "tails" has a greater probability than the other of showing. A coin which has a centre of gravity closer to the tails side than the heads side would be biased in that heads is more likely to show than tails. The size of coin can have an effect on the probability of heads and tails - during the Royal Institute Christmas lectures in the 1990s demonstrating probability a large version of the pound coin was made to be able to allow the audience to see it being tossed - on the broadcast (and tape) version it landed and stayed on its edge! showing the probability of heads = tails ≠ ½; the probability of heads = probability of tails, but they are actually slightly less than ½ as the coin could land on its edge and stay there - with a standard size coin, if it lands on its edge it takes very little for the centre of gravity to shift outside the base of the edge and for the coin to fall over, but with a very large similar coin (ie one scaled up [proportionally] in lengths) it can take quite a bit before the centre of gravity goes outside the base if it lands on its edge which forces it to fall over (plus there will be a "significant" rise in the centre of gravity to do so, thus favouring stability on an edge which does not exist in the standard, small, sized version of the coin).

No, two events are independent if the outcome of one does not affect the outcome of the other. They may or may not have the same probability. Flipping two coins, or rolling two dice, are independent. Drawing two cards, however, are dependent, because the removal of the first card affects the possible outcomes (probability) of the second card.

The probability of getting five heads out of 10 tosses is the same as the probablity of getting five tales out of ten tosses. One. It will happen. When this happens, you will get zero information. In other words, this is the expected result.

Since each event is independent (heads in one coin does not affect the probability of the other two coin flips), the multiplication rule applies: 1/2 x 1/2 x 1/2 = 1/8 or 0.125. So we can say the probability is 12.5%.

There is a fifty percent chance of the coin landing on "heads" each time it is flipped.However, flipping a coin 20 times virtually guarantees that it will land on "heads" at least once in that twenty times. (99.9999046325684 percent chance)You can see this by considering two coin flips. Here are the possibilities:Heads, heads.Heads, tails.Tails, tails.Tails, heads.You will note in the tossing of the coin twice that while each flip is fifty/fifty, that for the two flip series, there are three ways that it has heads come up at least once, and only one way in which heads does not come up.In other words, while it is a fifty percent chance for heads each time, it is a seventy five percent chance of seeing it be heads once if you are flipping twice.If you wish to know the odds of it not being heads in a twenty time flip, you would multiply .5 times .5 times .5...twenty times total. Or .5 to the twentieth power.That works out to a 99.9999046325684 percent chance of it coming up heads at least once in the twenty times of it being flipped.

Probability is the predictive chance that something will happen, such as the probability of heads coming up on a coin toss being 0.5.Consequence, on the other hand, is the result of some outcome, not necessarily related in any way to probability, but it could be. For instance, you may have a coin toss at the beginning of a football game. That is based on probability. The consequence, though, is the determination of who kicks off first.An example of non probabilistic consequence is the decision to swerve your car off of the road into a tree. The consequence of that action is that you will have a crash. Yes, its an extreme and unlikely example, but it shows that probability and consequence are two different things.

The probability of the coin coming up heads each time is 1/8; likewise for 3 tails. The probability of getting 2 heads and 1 tail (in any order) or 2 tails and 1 head, is 3/8. There are lots of other events whose probability can be calculated when a coin is tossed 3 times, but the question doesn't specify what event is to have its probability calculated.

(0.5)*(0.5)*(0.5) = 0.125 or 12.5% Please note the following in computing probabilities: Each toss is an independent event, the result of one toss does not affect the outcome of the next. The only outcomes are heads and tails. The probability of either outcome is 0.50 and does not change. Also note that many problems in real life are unlike coin flips, events are dependent on each other, outcomes are numerous, and not easily assigned probabilities. This does not necessarily make calculation of probabilities impossible, only more complicated.

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