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  probability


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A card is drawn from a deck of 52
cards. Find the probability of getting
a king or a heart or a red card.
letA, B, C be the events of getting
a king, a heart, a red card respectively.

denoting union with u and intersection with n

P(A) = 4 /52 [four kings]
P(B)=13/52 [thirteen hearts]
P(C)=26/52[twenty six red cards]

P(A n B)=1/52 [the king of hearts]
P(B n C)=13/52 [the thirteen hearts are all red]
P(C n A)=2/52 [2 kings among the red cards]

P(A n B n C )=1/52 [only the king of hearts is
common to all three events]

P[king OR heart OR redcard] =P[ A u B u C ]
=
P(A)+P(B)+P(C)-P(AnB)-P(BnC)-P(CnA)+P(AnBnC)
=4/52 +13/52 +26/52 -1/52 -13/52 -2/52 +1/52
=28/52=7/13
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problem 2
Verify the property that diagonals of a
rhombus bisect each other at right angles
for the rhombus with verticesA(0,5),B(-2,-2)
C(5,0),D(7,7)

diagonals are AC and BD
using midpoint formula({x1+x2}/2 , {y1+y2}/2)
midpoint of AC=({0+5}/2 , {5+0}/2)=(5/2 ,5/2)
midpoint of BD=({-2+7}/2, {-2+7}/2)=(5/2,5/2)

therefore
midpoint of AC = midpoint of BD
The diagonals bisect each other.

using slope formula
slope of AC=[{y2-y1}/{x2-x1}]=[(0-5)/(5-0)]=(-1)
slope of BD=[(7-(-2))/(7-(-2))]=1

product of the slopes of AC and BD =(-1)*1 = (-1)

AC and BD are at right angles to each other

Hence verified






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other questions and problems:


*integral calculus integration formula





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trigonometric identity and ratio of  certain standard angles

*list of differentiation formula     *integral calculus integration formula
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