Difference between revisions of "Science:MATH105 Probability/Lesson 1 DRV/1.01 Discrete Random Variables"

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==Probability==
 
==Probability==
  
We are often interested in quantifying the '''probability''' that a certain outcome in an experiment occurs. The probability that a certain event occurs is a number between 0 and 1, and represents how likely that event occurs. A probability of 0 implies that the outcome ''cannot'' occur, whereas a probability of 1 implies that the outcome ''must'' occur.  Any value in the interval (0, 1) means that the outcome only occurs some of the time.  
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We are often interested in quantifying the '''probability''', ''p'', that a certain outcome in an experiment occurs. The probability that a certain event occurs is a number between 0 and 1 that represents how likely that event occurs. A probability of 0 implies that the outcome ''cannot'' occur, whereas a probability of 1 implies that the outcome ''must'' occur.  Any value in the interval (0, 1) means that the outcome only occurs some of the time.  
  
When there are is a list of possible events in the experiment, the
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===Rules of Probability===
  
==Tossing a Fair Coin Once==
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When there is a discrete list of probabilities that can occur, we use the notation ''p<sub>k</sub>'' to denote the probability that event ''k'' will occur.
  
If we toss a coin into the air, there are only two possible outcomes: it will land as either "heads" (H) or "tails" (T). If the coin is a "fair" coin, it is equally likely that the coin will land as tails or heads.
 
  
Suppose we toss a fair coin and record whether it lands as H or T. If we repeat this experiment many times, the coin will be H 50% of the time and T 50% of the time. And so we say that the probability that a fair coin when tossed will be heads is 0.5.  
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==Example: Tossing a Fair Coin Once==
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If we toss a coin into the air, there are only two possible outcomes: it will land as either "heads" (H) or "tails" (T). If the tossed coin is a "fair" coin, it is equally likely that the coin will land as tails or heads. In other words, there is a 50% chance that the coin will land heads, and a 50% chance that the coin will land as tails.  
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Using our notation for probability, we can assign
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*  ''p<sub>H</sub>'' to be the probability that the tossed coin will land as heads
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*  ''p<sub>T</sub>'' to be the probability that the tossed coin will land as tails
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Because there are two outcomes that are equally likely, we assign the probability of 0.5 to each of them.
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*  ''p<sub>H</sub>'' = 0.5
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*  ''p<sub>T</sub>'' = 0.5
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As required, the sum of the probabilities equals 1, and each probability is a number in the interval [0, 1].  
  
 
==Tossing a Fair Coin Twice==
 
==Tossing a Fair Coin Twice==
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Because the coin is fair, each outcome is equally likely to occur. So, we assign the probability of 0.25 to each of these outcomes.
 
Because the coin is fair, each outcome is equally likely to occur. So, we assign the probability of 0.25 to each of these outcomes.
  
==Notation==
 
 
The two rules of probability
 
 
# 0 ≤ p ≤ 1
 
# sum p = 1
 
 
The ....
 
 
[[Category:MATH105]]
 
[[Category:MATH105]]
 
[[Category:MATH105 Probability]]
 
[[Category:MATH105 Probability]]
 
[[Category:MATH105 Lesson 1]]
 
[[Category:MATH105 Lesson 1]]

Revision as of 13:12, 15 January 2012

Note to MATH 105 instructors: here my goal is to define discrete probability with an example"

Discrete vs Continuous Random Variables

to go on the "About" page", or elsewhere

In MATH 105, we deal with

  • discrete random variables, and
  • continuous random variables.

You will notice that certain definitions and concepts between them will be related.

These concepts are introduced assuming no prior knowledge of probability, but they are discussed with an assumption of integration. That is, probability is introduced as an application of of integration.

Probability

We are often interested in quantifying the probability, p, that a certain outcome in an experiment occurs. The probability that a certain event occurs is a number between 0 and 1 that represents how likely that event occurs. A probability of 0 implies that the outcome cannot occur, whereas a probability of 1 implies that the outcome must occur. Any value in the interval (0, 1) means that the outcome only occurs some of the time.

Rules of Probability

When there is a discrete list of probabilities that can occur, we use the notation pk to denote the probability that event k will occur.


Example: Tossing a Fair Coin Once

If we toss a coin into the air, there are only two possible outcomes: it will land as either "heads" (H) or "tails" (T). If the tossed coin is a "fair" coin, it is equally likely that the coin will land as tails or heads. In other words, there is a 50% chance that the coin will land heads, and a 50% chance that the coin will land as tails.

Using our notation for probability, we can assign

  • pH to be the probability that the tossed coin will land as heads
  • pT to be the probability that the tossed coin will land as tails

Because there are two outcomes that are equally likely, we assign the probability of 0.5 to each of them.

  • pH = 0.5
  • pT = 0.5

As required, the sum of the probabilities equals 1, and each probability is a number in the interval [0, 1].

Tossing a Fair Coin Twice

Similarly, if we toss a fair coin two times, there are four possible outcomes:

  • HH
  • HT
  • TH
  • TT

Because the coin is fair, each outcome is equally likely to occur. So, we assign the probability of 0.25 to each of these outcomes.