You roll a six sided die. Find the probability of each of the following scenarios. A) Rolling a 5 or a number greater…
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Question “You roll a six sided die. Find the probability of each of the following scenarios. A) Rolling a 5 or a number greater…”
You roll a six sided die. Find the probability of each of the
following scenarios.
A) Rolling a 5 or a number greater than 3
B) Rolling a number less than 4 or an even number
C) Rolling a 6 or an odd number.
A) P (5 or number > 3)=______
B) P (1 or 2 or 3 or 4 or 6)=________
C) P (6 or 1 or 3 or 5)=________
Type an intger or a simplified fraction…….
Answer
This problem uses the concepts of probability and sample area.
Probability of an event refers to the probability of it occurring in random experiments.
A sample space is the sum of all elements found in a sample.
A probability of an event is between 0 to 1. It must be positive. It can be expressed in fractions or decimals, between 0 to 1. It can also be expressed in percentages, from 1 to 100.
The probability of an event is calculated as follows:
P\left( A \right) = \frac\rmFavorable number of cases\rmTotal number of cases in the sample spaceThe addition rule can be described as:
Pleft[A cupB right] = Pleft[A right] + Pleft[B right] - Pleft[A capB right]
(A)
The space required to roll a six-sided die once is the sample space.
S = \left\ 1,2,3,4,5,6 \right\Every event of rolling a six-sided die is equally probable. This means that each number has the same probability of appearing on the face.
You can calculate the probability of rolling a 5, by:
\beginarrayc\\P\left( \rmRolling a 5 on a six sided die \right) = \frac\rmFavorable number of cases\rmTotal number of cases in the sample space \\\\ = \frac16\\\endarrayThe sample area for rolling a number greater that 3 is.
S = \left\ 4,5,6 \right\The probability of rolling a number higher than 3 is therefore,
\beginarrayc\\P\left( \rmRolling a number greater than 3 \right) = \frac\rmFavorable number of cases\rmTotal number of cases in the sample space \\\\ = \frac36\\\endarrayThe sample area for rolling a 5, or a greater number than 3, is.
S_1 = \left\ 5 \right\Probability of rolling a 5, or a higher than 3, is.
\beginarrayc\\P\left( \rmRolling a 5 and a number greater than 3 \right) = \frac\rmFavorable number of cases\rmTotal number of cases in the sample space \\\\ = \frac16\\\endarrayProbability of rolling a 5, or any number higher than 3, is
\beginarrayc\\P\left( 5 \cup \rma number greater than 3 \right) = \left[ \beginarrayc\\P\left( \rm5 \right) + P\left( \rma number greater than 3 \right)\\\\ - P\left( 5 \cap \rma number greater than 3 \right)\\\endarray \right]\\\\ = \frac16 + \frac36 - \frac16\\\\ = \frac36\\\\ = \frac12\\\endarray
(B)
The sample area for rolling a number less that 4 is.
S = \left\ 1,2,3 \right\There is a chance that you will roll a number lower than 4.
\beginarrayc\\P\left( \rmRolling a number less than 4 \right) = \frac\rmFavorable number of cases\rmTotal number of cases in the sample space \\\\ = \frac36\\\endarrayThe sample area for rolling an even number is
{S_1} = \left\{ {2,4,6} \right\}It is possible to roll an even number.
\begin{array}{c}\\P\left( {{\rm{Rolling an even number}}} \right) = \frac{{{\rm{Favorable number of cases}}}}{{{\rm{Total number of cases in the sample space }}}}\\\\ = \frac{3}{6}\\\end{array}The sample area for rolling a number less that 4 and an even number is.
{S_2} = \left\{ 2 \right\}There is a chance that you will roll a number lower than 4, and an even number.
\begin{array}{c}\\P\left( {{\rm{Rolling a number less than 4 and an even number}}} \right) = \frac{{{\rm{Favorable number of cases}}}}{{{\rm{Total number of cases in the sample space }}}}\\\\ = \frac{1}{6}\\\end{array}There is a good chance that you will roll a number lower than 4, or an even number.
\begin{array}{c}\\P\left( {4 \cup {\rm{an even number}}} \right) = P\left( {\rm{4}} \right) + P\left( {{\rm{an even number}}} \right) - P\left( {4 \cap {\rm{an even number}}} \right)\\\\ = \frac{3}{6} + \frac{3}{6} - \frac{1}{6}\\\\ = \frac{5}{6}\\\end{array}
(C)
Probability of rolling a 6, is:
\begin{array}{c}\\P\left( {{\rm{Rolling a 6 on a six sided die}}} \right) = \frac{{{\rm{Favorable number of cases}}}}{{{\rm{Total number of cases in the sample space }}}}\\\\ = \frac{1}{6}\\\end{array}The sample area for rolling an odd number is
S = \left\{ {1,3,5} \right\}It is possible to roll an odd number.
\begin{array}{c}\\P\left( {{\rm{Rolling an odd number}}} \right) = \frac{{{\rm{Favorable number of cases}}}}{{{\rm{Total number of cases in the sample space }}}}\\\\ = \frac{3}{6}\\\end{array}The sample space for rolling a 6, or an odd number is
{S_2} = \phiProbability of rolling a 6, or an odd number, is
\begin{array}{c}\\P\left( {{\rm{Rolling a 6 and an odd number}}} \right) = \frac{{{\rm{Favorable number of cases}}}}{{{\rm{Total number of cases in the sample space }}}}\\\\ = \frac{0}{6}\\\\ = 0\\\end{array}Probability of rolling a 6, or odd number, is:
Part A of \begin{array}{c}\\P\left( {6 \cup {\rm{an odd number}}} \right) = P\left( {\rm{6}} \right) + P\left( {{\rm{an odd number}}} \right) - P\left( {6 \cap {\rm{an odd number}}} \right)\\\\ = \frac{1}{6} + \frac{3}{6} - 0\\\\ = \frac{4}{6}\\\\ = \frac{2}{3}\\\end{array}Ans
CodeMathTag22 calculates the probability that you will roll a 5, or any number greater than 3, and gives you \frac{1}{2}.
Conclusion
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