הנה פורים הוא בחי' יוה"כ כי יוה"כ פירושו כמו פורים.
Shneur Zalman of Lyadi
Purim manifests the same essence as Yom Kippur because Yom Kippurim means "like Purim."
כְּבַקָּרַת רוֹעֶה עֶדְרוֹ מַעֲבִיר צֹאנוֹ תַּחַת שִׁבְטוֹ כֵּן תַּעֲבִיר וְתִסְפֹּר וְתִמְנֶה וְתִפְקֹד נֶפֶשׁ כָּל חָי וְתַחְתֹּךְ קִצְבָה לְכָל בְּרִיָּה וְתִכְתֹּב אֶת גְּזַר דִּינָם
As a shepherd herds his flock, directing his sheep to pass under his staff, so you shall pass, count, and record the souls of all living, and decree a limit to each persons days, and inscribe the decreee of their judgment.
Is this an image of the shepherd tithing there sheep? Line the sheep up, pass them under the staff, and every tenth sheep is given to the priests in Jerusalem. How do randomness and chance connect Yom Kippur and Purim?
לכן אמרו בזוהר הקדוש יום כפורים שהוא כמו פורים, כלומר שהארת יום כפור הוא מטוהר וחושבנא מכאן ולהלאה ... כי העיקר האורות יום הכפורים לקנות לב חדש לעבוד אותו ... כן בפורים מה שחלף ועבר אין.
For that reason, they said in the Zohar that Yom Kippurim is like Purim, meaning that the light that shines from Yom Kippur is about purification and calculation from here forward .... For the essence of Yom Kippur's light is the acquisition of a new heart to worship [God]. .... Similarly, on Purim, what is past is gone.
From: "Purim and Yom Kippur: An Odd Couple?" By Rabbi Eliyahu Safran on the OU website
The pur, the goral of Purim and the casting of lots on Yom Kippur both speak to man’s basic condition of vulnerability, insecurity, and fickleness. A cursory reading of the megillah would find that the entire Purim narrative is defined by chance, unreasonable, absurd, and irrational events. One day Jews are secure in Persia; the next they face annihilation. One day Mordechai faces execution; the next he is Prime Minister. Haman’s conspiracy against the Jews arises from nowhere, it is as fickle and uncertain as the lots he draws to determine when to carry out his horrible intent. “Purim, therefore,” the Rav explains, “epitomizes the instability, uncertainty, and vulnerability which characterize human life generally but particularly govern the destiny of the Jews. . . . It alerts the Jew to the sudden turns of fortune, lurking dangers, the fickleness of life, even as the goral itself seems to operate through blind chance.”
Despite our search for answers, we are often faced with no sensible answer. Why does man sin? Why do two brothers, twins perhaps, mature to be so different, one a tzadik, the other a rasha? What combination of temptations, lures of intoxicating pleasures, appeals of political and social ideologies, home background, moral strength or weakness, changing fortunes, pressing circumstances and, yes, chance, play out in our lives?
These complex and compelling variables do nothing so much as cry out our utter vulnerability; our foolish and wicked behavior in response to them acknowledged by God while at the same time forgiven by Him.
“It is because of this,” says Rabbi Soloveitchik, “that man can stand before the Heavenly Bar of Justice, hoping for compassion and forgiveness. Despite his free will and accountability for his deeds, man enters his plea before the Almighty, claiming that he is not the author and designer of the worldly pleasures that were too powerful for him to resist.”
The lures of our physical world are all part of the goral that is the dynamic of our existence.
Is it not the purpose of Yom Kippurim for us to come face to face with our fallibility in the face of this goral? Like Purim, Yom Kippurim engages the big goral. The unknowns, the fickleness and irrationality of life that play out on a communal level in the Purim narrative do the same in the everyday lives of each man, compelling us to seek God’s compassion and forgiveness on Yom Kippur.
Each and every day, we live personal “Purims” in which tumult, confusion, uncertainty, danger and temptation toss and turn us; every night is Yom Kippur, a time when our regret, sorrow, insecurity, need for understanding, sensitivity, compassion, and forgiveness bring us to our knees before the Almighty God.