Question

A two digit number is written at
random (digit at 10s place is non-
zero). The probability that the
number will be even but smaller
than 40 is​

Answer 1

$\LARGE\underline\mathfrak{QueStion:}$

ᴀ ᴛᴡᴏ ᴅɪɢɪᴛ ɴᴜᴍʙᴇʀ ɪs ᴡʀɪᴛᴛᴇɴ ᴀᴛ ʀᴀɴᴅᴏᴍ (ᴅɪɢɪᴛ ᴀᴛ 10s ᴘʟᴀᴄᴇ ɪs ɴᴏɴ ᴢᴇʀᴏ).ᴛʜᴇ ᴘʀᴏʙᴀʙɪʟɪᴛʏ ᴛʜᴀᴛ ᴛʜᴇ ɴᴜᴍʙᴇʀ ᴡɪʟʟ ʙᴇ ᴇᴠᴇɴ ʙɪᴛ sᴍᴀʟʟᴇʀ ᴛʜᴀɴ 40 ɪs?

$\LARGE\underline\mathfrak{AnSwEr:}$

ᴡᴇ ᴀʀᴇ ɢɪᴠᴇɴ ᴛʜᴀᴛ ᴛʜᴇʀᴇ ᴀʀᴇ ᴛᴏᴛᴀʟ 90 ɴᴜᴍʙᴇʀs,

[ʜᴏᴡ :- 9 ᴘᴏssɪʙʟᴇ ᴅɪɢɪᴛs ɪɴ sᴇᴄᴏɴᴅ sʟᴏᴛ × 10 ᴅɪɢɪᴛs ɪɴ sʟᴏᴛ = ᴛᴏᴛᴀʟ 90 ɴᴜᴍʙᴇʀs]

ᴀɴᴅ ᴡᴇ ʜᴀᴠᴇ ᴛᴏ ғɪɴᴅ ᴛʜᴇ ᴘʀᴏʙᴀʙɪʟɪᴛʏ ᴏғ ᴀ ᴛᴡᴏ ᴅɪɢɪᴛ ɴᴜᴍʙᴇʀ ᴛʜᴀᴛ ɪs ᴄʜᴏsᴇɴ ᴀᴛ ʀᴀɴᴅᴏᴍ

ᴀʟsᴏ,

ᴛʜᴇ ᴛʜᴇ ᴛᴇɴs ᴘʟᴀᴄᴇ ᴅɪɢɪᴛ ᴏғ ᴛʜᴀᴛ ɴᴜᴍʙᴇʀ ɪs ɴᴏɴ ᴢᴇʀᴏ[ ᴛᴇɴ's ᴘʟᴀᴄᴇ ᴅɪɢɪᴛ ≠ 0 ]

sᴏ ʟᴇᴛ's ʙᴇɢɪɴ;

$\LARGE\underline\mathfrak{Step \: by \: step \: explanation:}$

ғᴏʀᴍᴜʟᴀ ᴜsᴇᴅ;

${\boxed{\sf \orange{ P(E) \: = \: \frac{ No. \: of \: favourable \: outcomes}{No. \: of \: total \: outcomes} }}}$

ɴᴏᴡ,ᴛᴏᴛᴀʟ ᴏᴜᴛᴄᴏᴍᴇs:-

ᴡᴇ ʜᴀᴠᴇ ᴛᴏ ғɪɴᴅ ᴘʀᴏʙᴀʙɪʟɪᴛʏ ғᴏʀ 90ᴏᴜᴛᴄᴏᴍᴇs ᴀs ɴᴜᴍʙᴇʀs,sᴏ ᴡᴇ ᴛᴀᴋᴇ

ɴᴏ. ᴏғ ᴛᴏᴛᴀʟ ᴏᴜᴛᴄᴏᴍᴇs = 90

ᴀʟsᴏ,ᴘᴏssɪʙʟᴇ ᴏᴜᴛᴄᴏᴍᴇs :-

ᴛʜᴇ ᴘᴏɪɴᴛ ᴛʜᴀᴛ sʜᴏᴜʟᴅ ʙᴇ ɴᴏᴛɪᴄᴇᴅ ʜᴇʀᴇ ;

ᴡᴇ ʜᴀᴠᴇ ᴛᴏ ғɪɴᴅ ᴘʀᴏʙᴀʙɪʟɪᴛʏ ᴏғ ɴᴜᴍʙᴇʀ ʟᴇss ᴛʜᴀɴ 40,ᴀᴍᴏɴɢ 40 ɴᴜᴍʙᴇʀs ᴛʜᴇʀᴇ ᴀʀᴇ ᴏɴᴇ's ᴀs ᴡᴇʟʟ ᴀs ᴛᴇɴ's ᴘʟᴀᴄᴇ ᴅɪɢɪᴛs ɴᴜᴍʙᴇʀs

sᴏ,ᴡᴇ ᴛᴀᴋᴇ ᴅɪғғᴇʀᴇɴᴄᴇ ᴏғ ᴀʟʟ ᴛʜᴇsᴇ ᴀɴᴅ ᴡᴇ ɢᴇᴛ

$\mapsto$ $\normalsize\tt\ 40 \: - \: 9 \: - \: 1 \: = \: 30$

ᴇᴠᴇɴ ᴏᴜᴛᴄᴏᴍᴇs;

$\mapsto$ $\normalsize\tt\frac{30}{2} \: = \: 15$

ɴᴏᴡ;

ɴᴜᴍʙᴇʀ ᴏғ ᴘᴏssɪʙʟᴇ ᴏᴜᴛᴄᴏᴍᴇs = 15

ᴀʟsᴏ,

ʜᴇʀᴇ ᴡᴇ ᴛᴀᴋᴇ "ᴇ" ᴀs ᴇᴠᴇɴᴛ ᴀɴᴅ ᴛʜᴇ

ᴇᴠᴇɴᴛ = ᴄʜᴏᴏsɪɴɢ ᴏғ ᴀ ᴛᴡᴏ ᴅɪɢɪᴛ ᴀᴛ ʀᴀɴᴅᴏᴍ.

ʏᴏᴜʀ ᴀɴsᴡᴇʀ,ᴀᴄᴄᴏʀᴅɪɴɢ ᴛᴏ ɢɪᴠᴇɴ ;

$\large\tt\ P(E) \: = \: \frac{15}{90} \: = \: \frac{1}{2}$

${\boxed{\sf \blue{P(E) \: = \: \frac{1}{6} }}}$

Answer 2

Step-by-step explanation:

ᴀ ᴛᴡᴏ ᴅɪɢɪᴛ ɴᴜᴍʙᴇʀ ɪs ᴡʀɪᴛᴛᴇɴ ᴀᴛ ʀᴀɴᴅᴏᴍ (ᴅɪɢɪᴛ ᴀᴛ 10s ᴘʟᴀᴄᴇ ɪs ɴᴏɴ ᴢᴇʀᴏ).ᴛʜᴇ ᴘʀᴏʙᴀʙɪʟɪᴛʏ ᴛʜᴀᴛ ᴛʜᴇ ɴᴜᴍʙᴇʀ ᴡɪʟʟ ʙᴇ ᴇᴠᴇɴ ʙɪᴛ sᴍᴀʟʟᴇʀ ᴛʜᴀɴ 40 ɪs?

\LARGE\underline\mathfrak{AnSwEr:}

AnSwEr:

ᴡᴇ ᴀʀᴇ ɢɪᴠᴇɴ ᴛʜᴀᴛ ᴛʜᴇʀᴇ ᴀʀᴇ ᴛᴏᴛᴀʟ 90 ɴᴜᴍʙᴇʀs,

[ʜᴏᴡ :- 9 ᴘᴏssɪʙʟᴇ ᴅɪɢɪᴛs ɪɴ sᴇᴄᴏɴᴅ sʟᴏᴛ × 10 ᴅɪɢɪᴛs ɪɴ sʟᴏᴛ = ᴛᴏᴛᴀʟ 90 ɴᴜᴍʙᴇʀs]

ᴀɴᴅ ᴡᴇ ʜᴀᴠᴇ ᴛᴏ ғɪɴᴅ ᴛʜᴇ ᴘʀᴏʙᴀʙɪʟɪᴛʏ ᴏғ ᴀ ᴛᴡᴏ ᴅɪɢɪᴛ ɴᴜᴍʙᴇʀ ᴛʜᴀᴛ ɪs ᴄʜᴏsᴇɴ ᴀᴛ ʀᴀɴᴅᴏᴍ

ᴀʟsᴏ,

ᴛʜᴇ ᴛʜᴇ ᴛᴇɴs ᴘʟᴀᴄᴇ ᴅɪɢɪᴛ ᴏғ ᴛʜᴀᴛ ɴᴜᴍʙᴇʀ ɪs ɴᴏɴ ᴢᴇʀᴏ[ ᴛᴇɴ's ᴘʟᴀᴄᴇ ᴅɪɢɪᴛ ≠ 0 ]

sᴏ ʟᴇᴛ's ʙᴇɢɪɴ;

\LARGE\underline\mathfrak{Step \: by \: step \: explanation:}

Stepbystepexplanation:

ғᴏʀᴍᴜʟᴀ ᴜsᴇᴅ;

{\boxed{\sf \orange{ P(E) \: = \: \frac{ No. \: of \: favourable \: outcomes}{No. \: of \: total \: outcomes} }}}

P(E)=

No.oftotaloutcomes

No.offavourableoutcomes

ɴᴏᴡ,ᴛᴏᴛᴀʟ ᴏᴜᴛᴄᴏᴍᴇs:-

ᴡᴇ ʜᴀᴠᴇ ᴛᴏ ғɪɴᴅ ᴘʀᴏʙᴀʙɪʟɪᴛʏ ғᴏʀ 90ᴏᴜᴛᴄᴏᴍᴇs ᴀs ɴᴜᴍʙᴇʀs,sᴏ ᴡᴇ ᴛᴀᴋᴇ

ɴᴏ. ᴏғ ᴛᴏᴛᴀʟ ᴏᴜᴛᴄᴏᴍᴇs = 90

ᴀʟsᴏ,ᴘᴏssɪʙʟᴇ ᴏᴜᴛᴄᴏᴍᴇs :-

ᴛʜᴇ ᴘᴏɪɴᴛ ᴛʜᴀᴛ sʜᴏᴜʟᴅ ʙᴇ ɴᴏᴛɪᴄᴇᴅ ʜᴇʀᴇ ;

ᴡᴇ ʜᴀᴠᴇ ᴛᴏ ғɪɴᴅ ᴘʀᴏʙᴀʙɪʟɪᴛʏ ᴏғ ɴᴜᴍʙᴇʀ ʟᴇss ᴛʜᴀɴ 40,ᴀᴍᴏɴɢ 40 ɴᴜᴍʙᴇʀs ᴛʜᴇʀᴇ ᴀʀᴇ ᴏɴᴇ's ᴀs ᴡᴇʟʟ ᴀs ᴛᴇɴ's ᴘʟᴀᴄᴇ ᴅɪɢɪᴛs ɴᴜᴍʙᴇʀs

sᴏ,ᴡᴇ ᴛᴀᴋᴇ ᴅɪғғᴇʀᴇɴᴄᴇ ᴏғ ᴀʟʟ ᴛʜᴇsᴇ ᴀɴᴅ ᴡᴇ ɢᴇᴛ

\mapsto↦ \normalsize\tt\ 40 \: - \: 9 \: - \: 1 \: = \: 30 40−9−1=30

ᴇᴠᴇɴ ᴏᴜᴛᴄᴏᴍᴇs;

\mapsto↦ \normalsize\tt\frac{30}{2} \: = \: 15

2

30

=15

ɴᴏᴡ;

ɴᴜᴍʙᴇʀ ᴏғ ᴘᴏssɪʙʟᴇ ᴏᴜᴛᴄᴏᴍᴇs = 15

ᴀʟsᴏ,

ʜᴇʀᴇ ᴡᴇ ᴛᴀᴋᴇ "ᴇ" ᴀs ᴇᴠᴇɴᴛ ᴀɴᴅ ᴛʜᴇ

ᴇᴠᴇɴᴛ = ᴄʜᴏᴏsɪɴɢ ᴏғ ᴀ ᴛᴡᴏ ᴅɪɢɪᴛ ᴀᴛ ʀᴀɴᴅᴏᴍ.

ʏᴏᴜʀ ᴀɴsᴡᴇʀ,ᴀᴄᴄᴏʀᴅɪɴɢ ᴛᴏ ɢɪᴠᴇɴ ;

\large\tt\ P(E) \: = \: \frac{15}{90} \: = \: \frac{1}{2} P(E)=

90

15

=

2

1