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Alex Ar [27]
3 years ago
9

A die with six faces has the number 1 painted on three of its faces, the number 2 painted on 2 of its faces, and number 3 on one

face. Assume each face is equally likely to come up.
a) Find a sample space.
b) Find P(odd number).
c) If die were loaded so that the face with 3 on it were twice as likely to come up as the other 5 faces, would this change the sample space?
d) If die were loaded so that the face with 3 on it were twice as likely to come up as the other 5 faces, would this change the value of P(odd number)?
Mathematics
1 answer:
salantis [7]3 years ago
7 0

Answer:

a) S = {1, 2, 3}

b) P(odd number) = \frac{2}{3}

c) No

d) Yes

Step-by-step explanation:

a) The sample space is the set of all possible outcomes. By definition, the elements of a set should not be repeated. Hence, the sample space S = {1, 2, 3}

However, the sample is not equiprobable because each element has different probabilities.

b) P(odd number) = \frac{number of odd digits}{number of faces}=\frac{4}{6}=\frac{2}{3}

Note that the odd numbers are 1 (on three faces) and 3 (on one face).

c) The fact the die has been biased does not change the possible outcomes. It only changes the probability of getting any given number.

d) Because the 3-face has been loaded, this probability changes. In fact, it is calculated thus:

Let's assume the probability for 1 or 2 is x. Then that of 3 is 2x(because it is twice the others). The sum of probabilities must be 1.

x+x+x+x+x+2x=1

7x=1

x=\frac{1}{7}

P(odd number) = 3\timesProb(1) + Prob(3)

= 3\times\frac{1}{7}+\frac{2}{7} = \frac{5}{7}

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<h3>Answer: No, there isn't enough ribbon</h3>

===========================================================

Use this formula

a & b/c = (a*c+b)/c

to convert from a mixed number to an improper fraction

So,

a & b/c = (a*c+b)/c

3 & 3/4 = (3*4+3)/4

3 & 3/4 = 15/4

---------------------------------------------------

Next, we add the fractions 15/4 and 5/6. The denominators aren't the same, so we can't add right away. We need to get the denominators to the LCD.

In this case, the LCD is 4*6 = 24

  • The fraction 15/4 becomes 90/24 after multiplying top and bottom by 6.
  • The fraction 5/6 becomes 20/24 after multiplying top and bottom by 4

Now we can add the fractions:

90/24+20/24 = (90+20)/24 = 110/24

---------------------------------------------------

So far, we've added the 3&3/4 portion to the 5/6 portion to get the result 110/24

The last thing to do is to add on the two 1/3 ft pieces, which means we're adding on 2/3 of a foot

2/3 = 16/24 after multiplying top and bottom by 8

Add this to the result of the last section

110/24+16/24 = (110+16)/24 = 126/24

---------------------------------------------------

From here, we convert the improper fraction 126/24 to decimal form

Using a calculator, you should find that 126/24 = 5.25

The result is larger than 5, which means she used too much ribbon. There won't be enough ribbon to do everything she wants to do.

---------------------------------------------------

The shortcut we can take is to type the following into the calculator

3+3/4+5/6+2/3

doing so should lead to 5.25

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