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chapter5small - Section 5.1 How Can Probability Quantify...

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Unformatted text preview: Section 5.1 How Can Probability Quantify Randomness? May 11, 2009 1 5 Probability in Our Daily Lives 5.1 How Can Probability Quantify Randomness? We will understand what the following statements mean: 1. The probability/chance of rain tomorrow is 20%. 2. The probability of inheriting Huntingtons disease, if exactly one parent has the disease, is 50%. 3. If two carriers of cystic fibrosis have a child, the child has a 25% chance of having the disease. 4. Only one in every 1,000,000 individuals has the blood type found at the crime scene. If an (innocent) individual is selected at random, the probability that she/he has DNA which match the blood at the crime scene is 1 out of 1,000,000. 5. Suppose we are interested in knowing if the new drug works better than the old drug. In a sample of 200 people, suppose 105 favor the new drug. Do we conclude the new drug is better? This is equivalent to tossing a coin. Now suppose 120? Now suppose 190? In sample survey , the sample is known and the population is unknown. We use the sample (and the statistics ) to make inferences about the population. Example: Poll 1000 people to learn about their income, for the sake of inferring about the incomes of the entire population. Poll 1000 people to learn about their income, for the sake of inferring about the incomes of the entire population. However, in probability , the population is known, and we discuss properties of the sample based on probability. Section 5.1 How Can Probability Quantify Randomness? May 11, 2009 2 Example: Use the above Huntingtons example. Use the above Huntingtons example. Definition: A probability is a number between 0 and 1, and reflects the likeli-hood of occurrence of some outcome. Example: Suppose a coin is fair . The probability of heads is 0.5, or P(heads)=0.5. The probability of heads is 0.5, or P(heads)=0.5. Example: Suppose a club has 6 females and 4 males. The relative frequency of club members who are female is 0.6. If a club member is randomly selected, then P(female)=0.6. The relative frequency of club members who are female is 0.6. If a club member is randomly selected, then P(female)=0.6. What does it mean when we say that P(heads)=0.5 for a coin? Alternative question: How do we find P(heads) for a coin? Here, we are using the concept of long run proportion to define probability. Here, we are using the concept of long run proportion to define probability. The graphs below represent the sample proportion of heads, in tosses of a fair coin, for a large number of tosses. Section 5.1 How Can Probability Quantify Randomness? May 11, 2009 3 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 0.00.10.20.30.40.50.6 P r o b a b i l i t y o f h e a d s i s 0 . 5 N u m b e r o f c o i n t o s s e s sample proportion of heads 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 0.50.60.70.80.91.0 P r o b a b i l i t y o f h e a d s i s 0 . 5 N u m b e r o f c o i n t o s s e s 0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0 0.4 0.6 0.8 1.0 P r o b a b i l i t y...
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chapter5small - Section 5.1 How Can Probability Quantify...

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