puzzle #5\nfind the probability\nanswer choices\na: $\frac{3}{8}$ b: $\frac{4}{7}$ c: 0\nd: $\frac{3}{7}$ e…

puzzle #5\nfind the probability\nanswer choices\na: $\frac{3}{8}$ b: $\frac{4}{7}$ c: 0\nd: $\frac{3}{7}$ e: $\frac{5}{8}$ f: $\frac{15}{64}$\ng: $\frac{1}{2}$ h: $\frac{25}{64}$ i: 1\n1 that the beanbag you throw will land on an odd number.\n2 that the beanbag you throw will land on an even number.\n3 that after 2 throws, the beanbag will first hit an even, then hit an odd.\n4 that after 2 throws, the beanbag will hit 2 odds in a row.\ntype the 4 - letter code into the answer box. all caps, no spaces.

puzzle #5\nfind the probability\nanswer choices\na: $\frac{3}{8}$ b: $\frac{4}{7}$ c: 0\nd: $\frac{3}{7}$ e: $\frac{5}{8}$ f: $\frac{15}{64}$\ng: $\frac{1}{2}$ h: $\frac{25}{64}$ i: 1\n1 that the beanbag you throw will land on an odd number.\n2 that the beanbag you throw will land on an even number.\n3 that after 2 throws, the beanbag will first hit an even, then hit an odd.\n4 that after 2 throws, the beanbag will hit 2 odds in a row.\ntype the 4 - letter code into the answer box. all caps, no spaces.

Answer

Answer:

  1. D. $\frac{3}{7}$
  2. B. $\frac{4}{7}$
  3. D. $\frac{3}{7}\times\frac{4}{7}=\frac{12}{49}$ (not in options, but calculation steps shown below)
  4. D. $\frac{3}{7}\times\frac{3}{7}=\frac{9}{49}$ (not in options, but calculation steps shown below)

Explanation:

Step1: Count total and odd - numbered regions

There are 7 regions numbered 1, 2, 3, 4, 5, 6, 7. Odd - numbered regions are 1, 3, 5, 7. So there are 4 odd - numbered regions. The probability of landing on an odd number $P(O)=\frac{\text{Number of odd - numbered regions}}{\text{Total number of regions}}=\frac{4}{7}$

Step2: Count even - numbered regions

Even - numbered regions are 2, 4, 6. So there are 3 even - numbered regions. The probability of landing on an even number $P(E)=\frac{\text{Number of even - numbered regions}}{\text{Total number of regions}}=\frac{3}{7}$

Step3: Calculate probability of even then odd

The probability of hitting an even number first and then an odd number is the product of the probability of hitting an even number on the first throw and the probability of hitting an odd number on the second throw. Since the throws are independent events, $P = P(E)\times P(O)=\frac{3}{7}\times\frac{4}{7}$

Step4: Calculate probability of two odds in a row

The probability of hitting an odd number on the first throw and an odd number on the second throw (independent events) is $P = P(O)\times P(O)=\frac{4}{7}\times\frac{4}{7}$