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There are 49 products in the box, from which only 6 are high quality. What is ...
Jan 31, 2024
There are 49 products in the box, from which only 6 are high quality. What is the probability of pulling out 6 random products from the box and having at least four of them high quality ?
Solution by Steps
step 1
Identify the total number of products and the number of high-quality products
step 2
Calculate the probability of success for one trial, which is the probability of selecting a high-quality product
step 3
Use the binomial probability formula to calculate the probability of getting exactly 4, 5, or 6 high-quality products in 6 trials
step 4
Add the probabilities of getting exactly 4, 5, and 6 high-quality products to find the total probability of getting at least 4 high-quality products
Answer
0.00274522
Key Concept
Binomial Probability Distribution
Explanation
The probability of getting at least 4 high-quality products out of 6 is calculated using the binomial probability formula, considering the number of successes (high-quality products) and the number of trials (products selected).
There are 60 chemical flasks in the laboratory, 6 of which are incorrectly labeled. What is the chance that if we randomly choose 5 flasks, exactly 3 of them will be labeled correctly ?
Solution by Steps
step 1
Calculate the number of ways to choose 3 correctly labeled flasks out of the 54 that are correctly labeled: (543)\binom{54}{3}
step 2
Calculate the number of ways to choose 2 incorrectly labeled flasks out of the 6 that are incorrectly labeled: (62)\binom{6}{2}
step 3
Calculate the total number of ways to choose 5 flasks out of 60 without any restriction: (605)\binom{60}{5}
step 4
Calculate the probability of choosing exactly 3 correctly labeled flasks and 2 incorrectly labeled flasks: (543)⋅(62)(605)\frac{\binom{54}{3} \cdot \binom{6}{2}}{\binom{60}{5}}
step 5
Simplify the expression to find the probability: (543)⋅(62)(605)=54⋅53⋅523⋅2⋅1⋅6⋅52⋅1⋅1(605)\frac{\binom{54}{3} \cdot \binom{6}{2}}{\binom{60}{5}} = \frac{54 \cdot 53 \cdot 52}{3 \cdot 2 \cdot 1} \cdot \frac{6 \cdot 5}{2 \cdot 1} \cdot \frac{1}{\binom{60}{5}}
step 6
Simplify further using the given asksia-ll calculation list: 60⋅59⋅58⋅57⋅565⋅4⋅3⋅2⋅1⋅15⋅201⋅1(605)\frac{60 \cdot 59 \cdot 58 \cdot 57 \cdot 56}{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1} \cdot \frac{15 \cdot 20}{1} \cdot \frac{1}{\binom{60}{5}}
step 7
Cancel out the (605)\binom{60}{5} terms: 15⋅201\frac{15 \cdot 20}{1}
step 8
Compute the final probability: 15⋅20=30015 \cdot 20 = 300
Answer
300
Key Concept
Probability of a Combined Event
Explanation
The probability of choosing exactly 3 correctly labeled flasks and 2 incorrectly labeled flasks is found by dividing the number of ways to choose such a combination by the total number of ways to choose any 5 flasks.
XA=BA=(2amp;5amp;76amp;3amp;45amp;−2amp;−3),B=(−1amp;1amp;00amp;1amp;11amp;0amp;−1)\begin{array}{l}X A=B \\ A=\left(\begin{array}{ccc}2 & 5 & 7 \\ 6 & 3 & 4 \\ 5 & -2 & -3\end{array}\right), B=\left(\begin{array}{ccc}-1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & -1\end{array}\right)\end{array}
Solution by Steps
step 1
To solve the matrix equation XA=BXA = B, we need to find matrix XX such that when it is multiplied by matrix AA, it yields matrix BB
step 2
According to the asksia-ll calculator, the equation has no solutions. This implies that matrix AA is singular or that the system of equations represented by the matrix equation is inconsistent
Answer
No solution exists for the given matrix equation.
Key Concept
Matrix Equation Solvability
Explanation
A matrix equation XA=BXA = B has a solution if matrix AA is non-singular (invertible). If the asksia-ll calculator indicates that no solutions exist, it means that AA is singular or the system is inconsistent, and therefore, there is no matrix XX that satisfies the equation.
Determine the relative position of the given straight lines, calculate the angle between them and find the intersection of the straight lines (if any exists) : е) p:{x=2−t;y=1+t;z=−2−t}\quad p:\{x=2-t ; y=1+t ; z=-2-t\} q:{x=1+s;y=s;z=5+s}q:\{x=1+s ; y=s ; z=5+s\}
Solution by Steps
step 1
Determine if the lines are parallel, intersecting, or skew by comparing the direction vectors
step 2
The direction vector for line p p is dp⃗=⟨−1,1,−1⟩ \vec{d_p} = \langle -1, 1, -1 \rangle and for line q q is dq⃗=⟨1,1,1⟩ \vec{d_q} = \langle 1, 1, 1 \rangle
step 3
Since dp⃗ \vec{d_p} is not a scalar multiple of dq⃗ \vec{d_q} , the lines are not parallel
step 4
To check if they intersect, set the parametric equations equal to each other and solve for t t and s s
step 5
Equating the x x -coordinates gives 2−t=1+s 2-t = 1+s which simplifies to s=1−t s = 1-t
step 6
Equating the y y -coordinates gives 1+t=s 1+t = s . Substituting s s from step 5 gives 1+t=1−t 1+t = 1-t , which simplifies to t=0 t = 0
step 7
Substituting t=0 t = 0 into s=1−t s = 1-t gives s=1 s = 1
step 8
Equating the z z -coordinates gives −2−t=5+s -2-t = 5+s . Substituting t=0 t = 0 and s=1 s = 1 gives −2=6 -2 = 6 , which is a contradiction
step 9
Since there is a contradiction, the lines do not intersect and are therefore skew
Answer
The lines are skew.
Key Concept
Determining the relationship between two lines in space
Explanation
Two lines in space can be parallel, intersecting, or skew. If their direction vectors are scalar multiples, they are parallel. If they can be made to intersect by finding common parametric values, they intersect. Otherwise, they are skew.
Solution by Steps
step 1
To find the angle between the lines, we use the dot product of their direction vectors
step 2
The dot product of dp⃗ \vec{d_p} and dq⃗ \vec{d_q} is (−1)(1)+(1)(1)+(−1)(1)=−1 (-1)(1) + (1)(1) + (-1)(1) = -1
step 3
The magnitude of dp⃗ \vec{d_p} is (−1)2+12+(−1)2=3 \sqrt{(-1)^2 + 1^2 + (-1)^2} = \sqrt{3} and the magnitude of dq⃗ \vec{d_q} is 12+12+12=3 \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}
step 4
The cosine of the angle θ \theta between the lines is given by cos⁡(θ)=dp⃗⋅dq⃗∣dp⃗∣∣dq⃗∣=−13 \cos(\theta) = \frac{\vec{d_p} \cdot \vec{d_q}}{|\vec{d_p}| |\vec{d_q}|} = \frac{-1}{3}
step 5
The angle θ \theta is then cos⁡−1(−13) \cos^{-1}\left(\frac{-1}{3}\right)
Answer
The angle between the lines is cos⁡−1(−13) \cos^{-1}\left(\frac{-1}{3}\right) .
Key Concept
Angle between two lines in space
Explanation
The angle between two lines can be found using the dot product of their direction vectors and the magnitudes of these vectors.
Solution by Steps
step 1
From the previous steps, we have determined that the lines are skew and therefore do not intersect
Answer
There is no intersection point as the lines are skew.
Key Concept
Intersection of two lines in space
Explanation
If two lines in space are skew, they do not intersect at any point.
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