Section 25-27 Probability and the Normal Curve

Question 1

Principles of Probability

Answer 1

PRINCIPLES OF PROBABILITY

1. MUTUALLY EXCLUSIVE events that are the RESULT OF CHANCE are INDEPENDENT of each other.
   
   • MUTUALLY EXCLUSIVE means that they are completely separate and have no bearing on each other.
     
     • Ex: If you roll a die and get a 6 (probability of 1/6), then what is the probability that if you roll that die again, a 6 will show up? A: 1/6, same as the first time and every time you roll it. That is true because multiple rolls of the die are MUTUALLY EXCLUSIVE – They have NOTHING to do with ONE ANOTHER!
2. SUMMING the probabilities of separate (mutually exclusive) outcomes in a SINGLE EVENT determines the probability that any of them will occur.
   
   • Rolling a die, the odds that a 6 will turn up is 1/6 because it is one side of a 6-sided die. The odds that a 4 will turn up is also 1/6. So the odds that EITHER a 6 OR a 4 turn up in a SINGLE ROLL of a 6-sided die is 1/6 + 1/6 = 2/6 ~ .33 ~ 33%
3. If something is CERTAIN to occur, the PROBABILITY of its occurrence is 1.00.
   
   • Ex: The probability that one of the numbers 1-6 turn up on the roll of a 6-sided die = 1.00 = 100%
4. If something is CERTAIN NOT to occur, the PROBABILITY of its nonoccurrence is 0.00.
   
   • Ex: The probability that the number 7 turns up on the roll of a STANDARD 6-sided die = 0.00 because there is no 7 to turn up.
5. Calculating the PRODUCT of the separate probabilities determines the probability of their SUCCESSIVE occurrence.
   
   • In #2 above, you take the SUM of MULTIPLE POSSIBLE MUTUALLY EXCLUSIVE OUTCOMES in a SINGLE EVENT to Determine the probability of EITHER a 6 OR a 4 turning up in a single roll of the die (1/6 + 1/6 = 2/6). It makes sense that one of two possible outcomes (a 6 OR a 4) from a roll of the die would be TWICE as likely as getting just one possible outcome (getting a 6)
   • HERE, we’re talking about something different. We’re talking about SUCCESSIVE rolls of a die. So what is the probability of rolling a 6 on the first roll and then ALSO rolling a 4 on the second roll? In this instance, you need to MULTIPLY the probabilities of each event.
     
     • Getting a 6 on the first roll is 1/6, while getting a 4 on the second roll is also 1/6. Multiplied, we get 1/6 * 1/6 = 1/36, which means it is MUCH less likely that we roll a die twice and get a 6 on the first roll and a 4 on the second roll than it is to get the 6 on a roll. It makes sense that being right TWICE is going to be much more difficult than being right just ONCE. But it’s important to notice that it’s not just twice as hard, it’s SIX TIMES as hard (with a 6-sided die) because you are essentially requiring something that has a 1/6 chance of happening (rolling a 6) before you can even consider the next part (rolling a 4 on the next roll), which is also only a 1/6 chance of happening. That is why it’s so difficult to guess the outcome of two CONSECUTIVE rolls of a die.
       
       • Imagine drawing a random playing card, returning it to the pile, shuffling, and then drawing a second card. The probability of guessing the outcome of that is 1/52 x 1/52 = 1/2,074 ~ .0004 ~ 0.04% chance. And that is why casinos make so much money.

Question 2

Probability and the Normal Curve

Answer 2

With a NORMAL DISTRIBUTION, we can use z-Scores to determine probabilities.

• Ex: Suppose we found that the distribution of math scores was normal, the mean was 50.00, and the standard deviation was 7.00.
• RECALL: z = (X-M) / S (z-score = (INPUT - AVERAGE) / Stan Dev)
• What is the probability of drawing an individual who has a score of 64 or higher at random from the population?
  
  1. Calculate the z-score z = (64 - 50.00) / 7.00 = 2.00
  2. lookup the z-score in Table 1 near the end of this book. OR, since the answer is a whole number z-score indicating +2.00 standard deviations from the mean, we can calculate the probability using the normal curve below.
     
     • There, we see that a score higher than 64 (higher than +2.00 standard deviations) includes the area to the RIGHT of 2σ.
       
       • That includes 2.14% + .14 = 2.28%. so the answer is that the probability of drawing an individual who got a 64 or higher randomly from this population is 2.28%.

This is referred to as a ONE-TAILED PROBABILITY because we asked the question about only the upper tail of the normal distribution-the right-hand tail of the distribution in the Figure below.

TWO-TAILED PROBABILITY – What is the probability of randomly drawing an individual with a z-score as high as 2.00 (or higher) OR as low as -2.00 (or lower)? Obviously, the odds of doing so are double those of drawing just one of these. Thus, the odds are 2 x .0228 = .0456, which is a little more than 4 in 100, or 4.56% probability.

• This is called a TWO-TAILED PROBABILITY because we are asking about the odds of drawing an individual at EITHER TAIL of the normal distribution

UNLIKELY EVENTS – The probabilities for both events described above represent unlikely events. In most sciences, any event that has a probability of occurrence of .05 (5%) or less is classified as unlikely to occur at random.

• A z-score of 1.96 has only a 2.5% chance of occurrence as a one-tailed probability. As a two-tailed probability, it has a 5.0% chance of occurrence (2 x 2.5% = 5.0%). Thus, an event with a z-score of 1.96 or greater or -1.96 or less (such as 1.97 or -1.97) is classified as an unlikely event.

Question 3

Percentile and Normal Curve

Answer 3

PERCENTILE RANK is the percentage of scores AT OR BELOW a given score. (See probability tab in Spreadsheet)